Quantum differential and difference equations for \mathrm{Hilb}^{n}(\mathbb{C}^2)
QQuantum differential and difference equationsfor Hilb n ( C ). Andrey Smirnov, [email protected] of North Carolina at Chapel HillSteklov Mathematical Institute of Russian Academy of Sciences
Abstract
We consider the quantum difference equation of the Hilbert schemeof points in C . This equation is the K-theoretic generalization ofthe quantum differential equation discovered by A. Okounkov andR. Pandharipande in [27]. We obtain two explicit descriptions for themonodromy of these equations - representation-theoretic and algebro-geometric. In the representation-theoretic description, the monodromyacts via certain explicit elements in the quantum toroidal algebra U (cid:126) ( (cid:98)(cid:98) gl ). In the algebro-geometric description, the monodromy fea-tures as transition matrices between the stable envelope bases in equiv-ariant K-theory and elliptic cohomology. Using the second approachwe identify the monodromy matrices for the differential equation withthe K-theoretic R -matrices of cyclic quiver varieties, which appear assubvarieties in the 3 D -mirror Hilbert scheme. Most of the results inthe paper are illustrated by explicit examples for cases n = 2 and n = 3 in the Appendix. Contents n ( C ) . . . . . . . . . . 31.2 Quantum difference equation for Hilb n ( C ) . . . . . . . . . . . 41.3 Monodromy of the quantum difference equation . . . . . . . . 51.4 Elliptic stable envelopes and mirror symmetry . . . . . . . . . 61.5 Quantum difference equation as qKZ . . . . . . . . . . . . . . 6 a r X i v : . [ m a t h . AG ] M a r .6 Connections with other topics in representation theory . . . . 7 n ( C ) z = 0 . . . . . . . . . . . . . . . . 102.4 Quantum connection and monodromy . . . . . . . . . . . . . . 112.5 Monodromy based at z = 1 . . . . . . . . . . . . . . . . . . . 122.6 Fundamental solution near z = ∞ . . . . . . . . . . . . . . . . 142.7 Transport of qde . . . . . . . . . . . . . . . . . . . . . . . . . 15 q -difference equation 16 gl . . . . . . . . . . 163.2 Wall-crossing operators . . . . . . . . . . . . . . . . . . . . . . 183.3 Monodromy operators . . . . . . . . . . . . . . . . . . . . . . 193.4 Basic properties of wall-crossing operators . . . . . . . . . . . 193.5 Quantum difference equation for Hilb n ( C ) . . . . . . . . . . 21 z = 0 . . . . . . . . . . . 224.2 Fundamental solution of QDE near z = ∞ . . . . . . . . . . . 234.3 Monodromy of QDE . . . . . . . . . . . . . . . . . . . . . . . 25 X . . . . . . . . . . . . . . . . . . . . . . 315.6 Monodromy from the elliptic stable envelope . . . . . . . . . . 32 B • w ( z ) . . . . . . . . . . . . . . . . . . 396.5 Gauss decomposition of B • w . . . . . . . . . . . . . . . . . . . 412 Wall-crossing operators as R-matrices 42 R -matrices . . . . . . . . . . . . . . . . . . . . . . 427.2 Wall crossing operators B w ( z ) as R -matrices of Y w . . . . . . 447.3 Monodromy operators B w as wall R -matrices . . . . . . . . . . 477.4 QDE as a non-minuscule qKZ equation . . . . . . . . . . . . 48 ∇ DT . . . . . . . . . . . . . . . . . . . . . . . . . 508.2 Cohomological limits of balanced functions . . . . . . . . . . . 528.3 Transport for s = 0 . . . . . . . . . . . . . . . . . . . . . . . 558.4 The fundamental group π ( P \ Sing , + ) . . . . . . . . . . . . 568.5 Fock representation of the fundamental group . . . . . . . . . 588.6 Relation in π ( P \ Sing , + ) . . . . . . . . . . . . . . . . . . . 59 X
62D Stable envelopes for Y w
64E Twisted R-matrices for Y w
66F Monodromy and wall-crossing operators 70G Representation of the fundamental group 71 n ( C ) In the study of Gromov-Witten/Donaldson-Thomas theories of threefoldsand representation theory of affine Yangians a certain first order differentialequation with remarkable properties emerges naturally. This equation fea-tures as the quantum differential equation (qde) in the equivariant quantumcohomology of the Hilbert scheme of points in the plane X = Hilb n ( C ) and3as first discovered and investigated by A. Okounkov and R. Pandharipandein [26, 27].The quantum differential equation describes a connection in a trivial bun-dle over P with fiber given by the equivariant cohomology H • T ( X ). Thesingularities of this connection are located at: Sing = { , ∞ , k √ , k = 2 , ..., n } \ { } ⊂ P where k √ k -th roots of 1. The most importantglobal properties of the qde are encoded in the homomorphism π ( P \ Sing , z ) → End( H • T ( X )) (1)provided by the monodromy of the corresponding connection. One of thegoals of this paper is to give a complete description of this homomorphismfor special choices of the based point z ∈ P .We recall the explicit form of the qde and its main properties in Section 2. n ( C ) In order to understand the monodromy of qde, we first consider its q -differencegeneralization: the quantum difference equation of the Hilbert scheme X isthe K-theoretic version of the quantum connection. This equation has thefollowing formΨ( zq ) = M ( z )Ψ( z ) , Ψ( z ) ∈ K T ( X ) , | q | < M ( z ) ∈ End( K T ( X )) was computed in Section 8 of [28]. Itis expressed via the action of the quantum toroidal algebra U (cid:126) ( (cid:98)(cid:98) gl ) on theequivariant K-theory of the Hilbert scheme K T ( X ) as follows. The algebra U (cid:126) ( (cid:98)(cid:98) gl ) is generated by elements α wk parameterized by w ∈ Q and k ∈ Z , seeSection 3.1 below. The action of U (cid:126) ( (cid:98)(cid:98) gl ) on the equivariant K -theories K T ( X )was constructed in [10]. For each rational number w = a/b we associate the wall-crossing operator B w ( z, q ) =: exp (cid:16) ∞ (cid:88) k =1 r wk α w − k α wk − z − bk q ak (cid:17) :4hich is an element in a completion of U (cid:126) ( (cid:98)(cid:98) gl ). The operator M ( z ) acts on K T ( X ) by: M ( z ) = O (1) −→ (cid:89) w ∈ Q − (cid:28) w< B w ( z, q ) (3)where O (1) denotes the operator of multiplication by the corresponding linebundle in K-theory. In the cohomological limit, which corresponds to sending q →
1, the q -difference equation (2) degenerates to the quantum differentialequation. It is well known that the q -difference equations are easier to work with thantheir differential limits. In particular, the multiplicative nature of q -differenceequations allows one to describe their fundamental solutions in terms of in-finite products. In Section 4 we use this idea to compute two fundamentalsolution matrices for (2), the first is holomorphic at z = 0 and the secondat z = ∞ . The monodromy of (2) is defined (according to Birkhoff) as thetransition matrix between these two fundamental solutions. From this wefind that the monodromy has the form of an ordered infinite product: Mon ( z ) ∼ −→ (cid:89) w ∈ Q B w ( z, q ) . (4)We explain our notation for the product over rational numbers later in Sec-tion 3.5. In cohomological limit q → Mon ( z ) we obtain description ofthe monodromy of the differentiation equation. In particular, we find thatthe operators B w := B w ( ∞ ,
1) =: exp (cid:16) ∞ (cid:88) k =1 r wk α w − k α wk (cid:17) : (5)describe the monodromy of the qde along the loop based at z = 0 which goesaround the singularity located at the root of unity z = e πiw , see Fig.4 inSection 8. As a result (Theorem 17) we obtain a complete description of thehomomorphism (1). 5 .4 Elliptic stable envelopes and mirror symmetry Thanks to its geometric origin, equation (2) plays distinguished role in theworld of q -difference equations. A geometric method for computing the mon-odromy for equations of this type, was recently proposed by M. Aganagic andA. Okounkov in [2, 24, 25]. In Section 5 we use their results to show that inthe basis of fixed points the monodromy operator admits a Gauss decompo-sition: Mon ( z ) = U ( z ) − L ( z ) (6)where U ( z ) and L ( z ) are certain upper and lower-triangular matrices, withmatrix elements given by the fixed points components of the stable envelopeclasses in the equivariant elliptic cohomology of X .Comparing (4) and (6) and using 3 D -mirror symmetry of elliptic stableenvelopes we obtain the following result (see Theorem 12 for precise state-ment): B w ( z, q ) = R − Y w ( zq − w ) (7)where R − Y w ( z ) is the K-theoretic R -matrix of certain subvariety Y w ⊂ X ! , inthe (also known as the symplectic dual) Hilbert scheme X ! . For arational number w = a/b the irreducible components of Y w are isomorphic toNakajima varieties associated with a cyclic quiver with b vertices, see Fig. 2.Form the representation-theoretic viewpoint, the R -matrix R − Y w ( zq − w ) corre-spond to the trigonometric R -matrix of the quantum toroidal algebra U (cid:126) ( (cid:98)(cid:98) gl b ).Comparing (7) with (5) we conclude that the monodromy of the qde alonga loop which goes around the singularity located at z = e πiw is described bythe K-theoretic R -matrix of the mirror variety Y w : B w = R − Y w := R − Y w ( ∞ )Finally, let us note that the wall-crossing operators B w ( z, q ) were computedin [28] using the machinery of Hopf algebras developed in [8], which is notgeometric in its nature. Now, we can view relation (7) as independent, purelygeometric definition of this operators. Using (7) we may rewrite the operator (3) as an ordered product of R -matrices. Thus, (2) takes a form similar to the quantum Knizhnik–Zamolodchikov quation , see (95) and Section 7.4 for discussion. We note, however, an impor-tant difference: the R -matrices entering the standard qKZ equations are all R -matrices of the same quantum group. In contrast, R − Y w ( z ) in (95) are the R -matrices associated with cyclic quivers with length b which vary with w .First, this suggests to view the equation (95) as a proper generalization ofqKZ equations to the case of toroidal quantum groups. Second, this suggestconsidering the the quantum difference and differential equations of X as thegeneralized qKZ equations associated with mirror variety X ! . From representation-theoretic perspective, the quantum differential equationfor the Hilbert scheme X describes the so called Casimir connection for theaffine Yangian Y (cid:126) ( (cid:98) gl ) [17]. The monodromy of the Casimir connection for theclassical symplectic resolutions X = T ∗ ( G/P ), where P ⊂ G is a parabolicsubgroup, describes the action of the quantum Weyl group associated with G [36, 35]. Thus, the monodromy operators B w can be viewed as generatorsof the “quantum Weyl group of the quantum toroidal algebra U (cid:126) ( (cid:98)(cid:98) gl )”. Thewall-crossing operators B w ( z, q ) with the parameter z “turned on” providethe dynamical version of this Weyl group, with z playing the role of thedynamical parameter. We refer to [8] for the introduction to the dynamicalWeyl groups.Important structures emerge the from the algebra A X = quantization of X known as Cherednik’s spherical DAHA for gl ( n ). Using quantization in primecharacteristic p (cid:29) D b Coh ( X ) [3]. This leads to acategorification of the monodromy operators B w . The categorification of thedynamical operators B w ( z, q ) remains to be understood in this approach.Perhaps (7), relating these operators to the K-theoretic R -matrices of thesymplectic dual variety X ! gives a hint in this direction.7 cknowledgments This paper emerged from the author’s attempt to write an “example” fora project currently developed in [14, 15] and we thank Yakov Kononov forcollaboration. We thank Hunter Dinkins for reading preliminary version ofthe paper and useful suggestions. We thank Andrei Okounkov from whomwe learned many of the ideas discussed here.The work is partially supported by the Russian Science Foundation undergrant 19-11-00062. n ( C ) In this paper X = Hilb n ( C ) - the Hilbert scheme of n points in C . Wedenote by T the two-dimensional complex torus acting on X via( x, y ) → ( (cid:15) x, (cid:15) y )where ( x, y ) denote the coordinates on C . The T -equivariant quantum coho-mology ring of X is computed in [26]. The corresponding quantum differentialequation is considered in [27]. This section is a brief overview of these results. Let us consider Q ( (cid:15) , (cid:15) )-vector space Fock = Q [ p , p , p , . . . ] ⊗ Z Q ( (cid:15) , (cid:15) ) . (8)Let us consider the Heisenberg algebra generated by α n , n ∈ Z \{ } satisfyingthe relations [ α n , α m ] = nδ n + m . This algebra acts on the Fock space via α n ( f ) = − n dfdp n , n > , − p − n f, n < . (9)8he vectors p µ = (cid:89) i p µ i (10)form a basis of the Fock space labeled by partitions µ = ( µ , µ , . . . ). Thereexist an isomorphism of Q ( (cid:15) , (cid:15) ) vector spaces ∞ (cid:77) n =0 H • T (Hilb n ( C )) loc = Fock (11)Under this isomorphism, the canonical basis of the torus fixed points in H • T (Hilb n ( C )) loc is identified with the basis of normalized Jack polynomials J λ in the Fock space. The summand in (11) corresponding to a fixed valuesof n is spanned by the Jack polynomials J λ with | λ | = n . Examples of Jackpolynomials can be found in Appendix A. Let us consider the following operator acting on the Fock space: m ( z ) = 12 (cid:88) k,l> ( (cid:15) (cid:15) α k + l α − k α − l + α − k − l α k α l )+( (cid:15) + (cid:15) ) ∞ (cid:88) k =1 k z k + 1 z k − α − k α k − (cid:15) + (cid:15) z + 1 z − | · | (12)where | · | maps p µ to | µ | p µ . The operator m (0) corresponds to the classicalmultiplication by the fist Chern class in equivariant cohomology. In partic-ular, it is diagonal in the basis of fixed points (Jack polynomials) with thefollowing eigenvalues: m (0)( J λ ) = c (0) λ J λ , c (0) λ = (cid:88) ( i,j ) ∈ λ ( i − (cid:15) + ( j − (cid:15) . (13)One also observes that m ( z − ) = − ( − l m ( z )( − l (14)where we denote the operator( − l : p µ → ( − l µ p µ . m ( ∞ )( J ∗ λ ) = c ( ∞ ) λ J ∗ λ , (15)where J ∗ λ = J λ | p i = − p i denote the Jack polynomials in variables − p i and c ( ∞ ) λ = − c (0) λ . (16)The main object of study in [27] is the differential equation (qde): ∇ ψ = 0 , ψ ∈ Fock , ∇ = z ddz − m ( z ) . (17)For a fixed value of n this equation has regular singularities located at Sing = { , ∞ , k √ , k = 2 , ..., n } \ { } ⊂ P (18)where k √ k -th roots of 1. Note that z = 1 is not a singularity of qde. z = 0 Let J be the matrix with λ -th column given by J λ . We assume that thecolumns of J are ordered by the standard dominance order on partitions . By(13) we have m (0) J = J c (0) where c (0) is the diagonal matrix with eigenvalues c (0) λ . Explicit examples ofmatrices J can be found in Appendix A. Remark 1.
Unless otherwise stated, all operators acting in the Fock spaceare considered in the basis (10). Thus, we sometimes refer to the operatorsas “matrices” without confusion. In this view, the matrix J is the transitionmatrix from the basis “Gromov-Witten” basis p λ to the “Donaldson-Thomas”basis J µ of the Fock space, see Appendix A.By basic theory of ordinary differential equations, the solutions of qde(17) in D = {| z | < } are described by the fundamental solution matrix ofthe form: ψ ( z ) = ψ reg ( z ) z c (0) , (19)10here z c (0) denotes the diagonal matrix with eigenvalues z c (0) λ , the matrix ψ reg ( z ) is holomorphic in D and is normalized by the condition ψ reg (0) = J (20)The coefficients in the Taylor expansion of ψ reg ( z ) at z = 0 are uniquelydetermined by equation (17) and “initial condition” (20). Remark 2.
By definition, the columns of the matrix ψ ( z ) form a basis ofsolutions of qde. Any other solution is a Q ( (cid:15) , (cid:15) )-linear combination of thesesolutions. This means that all other fundamental solution matrices of theqde are of the form ψ ( z ) A for some invertible A ∈ End(
Fock ). ∇ is a flat connection in a trivial bundle over P \ Sing with a fiber
Fock . Themonodromy of this connection defines a representation of the fundamentalgroup π ( P \ Sing , p ) based at a point p ∈ P \ Sing .In Section 8 we compute the monodromy for p = 0 + - a point “infinitesi-mally” close to 0 . In this case it is convenient to normalize the fundamentalsolution matrix by ψ DT ( z ) := ψ ( z ) Γ DT (21)where Γ DT denotes the diagonal matrix with eigenvalues(Γ DT ) λ,λ = (cid:89) w ∈ char T ( T λ X ) w + 1) . (22)and Γ( x ) stands for the standard Gamma function of x . In other words, the λ -th column of the fundamental solution matrix (19) is multiplied by theproduct of gamma functions (22).Let γ ∈ π ( P \ Sing , + ) and let us denote by ψ DT ( γ · z ) the solution ofthe qde which is obtained from ψ DT ( z ) via an analytic continuation alonga loop representing γ . The columns of the matrix ψ DT ( γ · z ) are solutions p = 0 as a base point.
11f the qde and thus are linear combinations of the columns of the originalfundamental solution matrix, see Remark 2. Thus ψ DT ( γ · z ) = ψ DT ( z ) · ∇ DT ( γ )for some matrix ∇ DT ( γ ) ∈ End(
Fock ). The map ∇ DT : γ (cid:55)→ ∇ DT ( γ )defines an antihomomorphism ∇ DT : π ( P \ Sing , + ) −→ End(
Fock ) , (23)which is called the monodromy of qde. Remark 3.
This map is an antihomomorphism because fundamental solu-tions transform in (23) by multiplication from the right, in particular: ψ DT ( γ w · γ w · z ) = ψ DT ( γ w · z ) ∇ DT ( γ w ) = ψ DT ( z ) ∇ DT ( γ w ) ∇ DT ( γ w )so that ∇ DT ( γ w · γ w ) = ∇ DT ( γ w ) · ∇ DT ( γ w ) . Thanks to the choice of normalization (22) the monodromy operatorshave good properties:
Theorem 1.
The matrix elements of matrices ∇ DT ( γ ) , γ ∈ π ( P \ Sing , + ) are rational functions in t = e πi(cid:15) , t = e πi(cid:15) . (24) Proof.
Apply logic of Theorem 3 in [27].In Section 8 we compute the monodromy matrices ∇ DT ( γ ) for a certainchoice of generators of π ( P \ Sing , + ). z = 1 In [27], instead of the fundamental group based at z = 0 + the nonsingularbased point z = 1 is considered. In this case it is convenient to work with12he fundamental solution matrix ψ GW ( z ) which is holomorphic near z = 1and is normalized by ψ GW (1) = Γ GW . (25)where Γ GW denotes the diagonal matrix with eigenvalues(Γ GW ) λ,λ = (cid:89) i g ( µ i , t ) g ( µ i , t ) , g ( x, t ) = x xt Γ( tx ) . For γ ∈ π ( P \ sing ,
1) this solution transforms as ψ GW ( γ · z ) = ψ GW ( z ) ∇ GW ( γ )which provides an antihomomorphism ∇ GW : π ( P \ Sing , −→ End(
Fock ) (26)With normalization (25) the monodromy matrices ∇ GW ( γ ) have good prop-erties: Theorem 2 (Theorem 3 in [27]) . Let γ ∈ π ( P \ Sing , , then the matrixelements of matrices ∇ GW ( γ ) are Laurent polynomials in t , t given by (24). The monodromies of ∇ DT and ∇ GW are related by the transport of qde from0 to 1, which is described by the following important result: Theorem 3 (Theorem 4 in [27]) . At z = 1 the solution (21) has the followingform: ψ DT (1) = Γ GW P where P is the matrix with λ -th column given by Macdonald polynomial P λ in Haiman’s normalization. Examples of Macdonald polynomials and matrices P λ can be found inAppendix B.Let γ ∈ π ( P \ sing , + ) and let γ (cid:48) = tγt − ∈ π ( P \ sing ,
1) where t isthe real path from 0 + to 1, then we have: Proposition 1.
The matrices ∇ DT ( γ ) and ∇ GW ( γ (cid:48) ) are related by: ∇ GW ( γ (cid:48) ) = P ∇ DT ( γ ) P − . (27) Proof.
By Theorem 3 and (25) we have ψ DT (1) = ψ GW (1) P thus the corresponding monodromies are conjugated by P .13 .6 Fundamental solution near z = ∞ Let J ∗ be the matrix with λ -th column given by the vector J ∗ λ . By (15) wehave m ( ∞ ) J ∗ = J ∗ c ( ∞ ) where c ( ∞ ) is the diagonal matrix with eigenvalues c ( ∞ ) λ . In D ∞ = | z | > ψ ∞ ( z ) = ψ reg ∞ ( z ) z c ( ∞ ) , (28)with ψ reg ∞ ( z ) holomorphic in D ∞ normalized by ψ reg ∞ ( ∞ ) = J ∗ . (29)The coefficients of the Taylor series expansion of ψ reg ∞ ( z ) at z = ∞ areuniquely determined by the qde (17) and initial condition (29). As abovewe also define ψ ∞ DT ( z ) := ψ ∞ ( z ) Γ DT . (30) Important : this time we assume that the columns of matrix ψ ∞ DT ( z ) areordered by opposite dominance order on partitions . Explicitly, this meansthe following: the dominance order and the opposite dominance order onpartitions are related by the transposition λ → λ (cid:48) . Let us consider thecorresponding matrix S λ,µ := δ λ,µ (cid:48) = · · · · · · · · · · · · · · · . Then, we assume that the initial conditions are given by ψ reg ∞ ( ∞ ) = J ∗ = ( − l J S and, respectively ψ ∞ DT ( z ) z − c ( ∞ ) (cid:12)(cid:12)(cid:12) z = ∞ = ( − l J Γ DT S . As we will see below, with this choice of normalization, the transports of theqde from z = 0 to z = ∞ have a natural Gauss decomposition.14 .7 Transport of qde Assume that we have analytic continuations of the fundamental solutions ψ DT ( z ) and ψ ∞ DT ( z ) to some larger domains. In this case we can comparethem at the points of { z ∈ C : | z | = 1 , z (cid:54)∈ Sing } . Explicitly, this is the set of points z = e πis for some s ∈ R . Let us considerthe transition matrix between two fundamental solutions:Tran DT ( s ) = ψ DT ( e πis ) − ψ ∞ DT ( e πis ) . (31)This operator describes the transport of the qde from z = 0 to z = ∞ alongthe line which intersects | z | = 1 at e πis . The transport depends only onhomotopy equivalence class of the path. This means that Tran DT ( s ) is apiecewise constant function of s , which changes value only when e πis hits asingularity. Theorem 4 (Section 4.6 [27]) . The transport of qde along the positive partof the real axis R + equals: Tran DT (0) = P − P ∗ . where P ∗ = ( − l P S . Proof.
Let ψ ∞ DT ( z ) be the solution (28). Theorem 3 together with (14) gives ψ ∞ DT (1) = Γ GW ( − l P S Thus, Tran DT (0) = ψ DT (1) − ψ ∞ DT (1) = P − ( − l P S . In Section 8 we generalize this result and describe Tran DT ( s ) for an arbitrary s ∈ R such that e − πis (cid:54)∈ Sing . 15
K-theoretic q -difference equation The quantum difference equation (QDE) is the K-theoretic generalization ofthe quantum differential equation (qde) in cohomology. For the Nakajimaquiver varieties these equations were computed in [28]. In particular, thecase of the Hilbert scheme Hilb n ( C ) is considered in detail in Section 8 of[28]. In this section we briefly review this construction. gl In this section we denote
Fock = Q [ p , p , p , . . . ] ⊗ Z Q ( t , t ) . (32)with t , t given by (24). There is an isomorphism of Q ( t , t )-vector spaces ∞ (cid:77) n =0 K T (Hilb n ( C )) loc = Fock (33)Under this isomorphism the K-theory classes O λ ∈ K T ( X ) of fixed points λ ∈ X T are mapped to Macdonald polynomials P λ ∈ Fock in Haiman’s nor-malization [13]. These polynomials form a basis of the Fock space, see Ap-pendix B for examples of P λ .Similarly to the qde in cohomology, the K-theoretic QDE for Hilb n ( C ) isdescribed via an action of certain algebra on Fock . The K-theoretic structureis, however, much richer. The Fock space (32) is a natural representation ofa quantum group U (cid:126) ( (cid:98)(cid:98) gl ), called quantum toroidal gl . The representationtheory of this algebra and its role in mathematical physics is an exceptionallyrich subject: we refer to [18, 4, 21, 32, 19, 9, 10] for a very incomplete list ofresearch by different groups.The algebra U (cid:126) ( (cid:98)(cid:98) gl ), has an explicit presentation in terms of generatorsand relations [32]: U (cid:126) ( (cid:98)(cid:98) gl ) = (cid:68) e ( n,m ) : ( n, m ) ∈ Z \ (0 , (cid:69) / relationsGiven a rational number w = ab with gcd( a, b ) = 1 one can consider thegenerators with “slope” w , see Fig. 1: α wk = e bk,ak , k ∈ Z \ { } . w ∈ Q ∪ {∞} the elements α wk generate the slope w Heisenbergsubalgebra H w ⊂ U (cid:126) ( (cid:98)(cid:98) gl ) subject to the following relations:[ α wi , α wj ] = δ i + j r ( w ) i , r ( w ) i = i ( (cid:126) kb − (cid:126) − kb )( t k − t − k )( t k − t − k )( (cid:126) k − (cid:126) − k )In particular, the slope w = 0 Heisenberg subalgebra α k = e ( k,
0) acts onthe Fock space by α m ( f ) = − p − m · f ( t m/ − t − m/ )( t m/ − t − m/ ) m < − m dfdp m m > w = ∞ -subalgebrais commutative (in this case b = 0). The elements α ∞ k = e (0 , k ) act diagonallyin the basis of fixed points P λ and can be identified with operators of multipli-cation by tautological bundles in the equivariant K-theory K T (Hilb n ( C )),see Section 8.1.5 of [28] for more details. The operators α ∞ k can also beidentified with so called Macdonald operators for gl ∞ .For a general slope w = ab the operators α wk act in (33) changing theweight by kb units α wk : K T (Hilb n ( C )) −→ K T (Hilb n − kb ( C )) . (34)In particular, α wk ( f ) = 0 for any f ∈ K T (Hilb n ( C )) with n < bk . For thisreason α w − k with k > creation operators and α wk as annihilation operators . The action of the operators α wk on the Fock spacefor general w and k is quite complicated. Still, it can be described explicitlyin the fixed point basis [20]. 17igure 1: The structure of the toroidal algebra and the Heisenberg subalgebraof slope w = 1 / For any w ∈ Q ∪ {∞} we define the wall-crossing operator B w ( z, q ) actingon the Fock space by B w ( z, q ) =: exp (cid:16) ∞ (cid:88) k =1 r wk − z − bk q ak α w − k α wk (cid:17) : (35)where a and b denote the numerator and the denominator of w and q isa formal complex parameter. The symbol :: denotes the normally orderedoperator. This means that in the Taylor expansion of the exponent (35) the annihilation operators α wk with k > creation operators α w − k .By (34) only finitely many terms in this expansion act non-trivially. Thusthe action of B w ( z, q ) for w ∈ Q is well defined.It is clear from (35) that the wall-crossing operators preserve the sum-mands in (33). By the same reason as above the operators B w ( z, q ) actnon-trivially (i.e., B w ( z, q ) (cid:54) = 1 ) on K T (Hilb n ( C )) only for w correspondingto the Farey sequence of order b : Walls n = (cid:110) w = ab ∈ Q : 1 ≤ | b | ≤ n (cid:111) ⊂ Q walls . From the explicit formulas for the action of the ∞ -slope Heisenberg algebra one finds that the action of B ∞ ( z, q ) is also welldefined, but we will not need it in this paper.Finally, we note that the matrix elements of B w ( z, q ) are rational func-tions in z and q . Using the action of U (cid:126) ( (cid:98)(cid:98) gl ) on the Fock space the matrices B w ( z, q ) can be computed very explicitly, see examples in Sections 8.3.7-8.3.8of [28]. Note that B w (0 , q ) = 1. We denote B w = B w ( ∞ , q ) =: exp (cid:16) ∞ (cid:88) k =1 r wk α w − k α wk (cid:17) : (36)We call B w the monodromy operators . The matrix elements of B w do notdepend on z or q . From the explicit formulas above one computes B =( t t ) n . It follows from Proposition 2 below that B w = ( t t ) n , w ∈ Z . In contrast, the monodromy operators B w for non-integral values of w arequite nontrivial, see the examples in Appendix G.We denote B w ( z ) := B w ( z, Lemma 1.
For s ∈ Q we have lim q → B w ( zq s , q ) = B w w > s w < s B w ( z ) s = w (37)In Section 8.4 we prove that the operator B w describes the monodromy of ∇ DT along the loop around the singularity of ∇ DT at z = e πiw , which explainsour choice of its name. Let us denote by L the operator of tensor multiplication by the line bundle O (1) in equivariant K-theory. In the basis of fixed points it is characterized19y the following eigenvalues L ( P λ ) = ( (cid:89) ( i,j ) ∈ λ t i − t j − ) P λ . (38) Proposition 2. L intertwines the action of the Heisenberg algebras H w and H w − on the Fock space: L − α wk = α w − k L − . In particular, it intertwines the wall crossing operators L − B w ( zq, q ) L = B w − ( z, q ) (39) and the monodromy operators L − B w ( z ) L = B w − ( z ) , L − B w L = B w − . Proof.
See [28], Section 8.
Proposition 3.
For arbitrary complex parameters a and b the wall-crossingoperators commute B w ( a, q ) B w ( b, q ) = B w ( b, q ) B w ( a, q ) , i.e., the coefficients of the Taylor expansion B w ( z ) = ∞ (cid:80) k =0 B w,k z k commute B w,i B w,j = B w,j B w,i Proof.
Follows directly from (35) and relations in H w .The following result describes the transformation of the wall-crossing op-erators and the quantum difference equation under z → z − . Proposition 4.B w ( z, q ) B − w = B − w B w ( z, q ) = B w ( z − , q − ) − in particular, at q = 1 B w ( z ) B − w = B − w B w ( z ) = B w ( z − ) − . Proof.
Follows directly from (35) and relations in H w .20 .5 Quantum difference equation for Hilb n ( C ) The quantum difference equation for Hilb n ( C ) (QDE) has the following formΨ( zq ) = M ( z )Ψ( z ) , Ψ( z ) ∈ Fock (40)where M ( z ) ∈ End(
Fock ) is given by M ( z ) = L −→ (cid:89) w ∈ [ − , B w ( z, q ) . (41)In the “classical limit” z = 0 the operator M ( z ) coincides with L : M (0) = L . (42)In (41) and throughout this paper we use the following conventions: −→ (cid:89) w ∈ I B w ( z, q )denotes the product over the rational numbers in an interval I ⊂ R orderedso that w increases from the left to the right. Similarly, ←− (cid:89) w ∈ I B w ( z, q )denotes the ordered product of operators with w increasing from right to left.If I ∩ Walls n is finite, as for instance in (41), then all but finitely many termsin these products act on the Fock space as identity operators. Thus, for fixed n and bounded I these products are finite and well defined. Example 1.
Let us assume that n = 3 then, for instance −→ (cid:89) w ∈ [0 , B w ( z ) = B ( z ) B ( z ) B ( z ) B ( z )or ←− (cid:89) w ∈ ( − , B w = B B B B B − B − B − . Solutions of QDE and monodromy
The multiplicative nature of q -difference equations allows us to construct fun-damental solutions and monodromy matrices as infinite products of the wallcrossing operators. For instance, in the case of zero-dimensional A ∞ quivervarieties we are dealing with the first order scalar q -difference equations andthe analysis of monodromy is elementary, see Section 5 of [7]. In this sectionapply the same logic for the Hilbert scheme Hilb n ( C ). z = 0 By (42) the operator M (0) is diagonal in the basis of fixed points, which inthe Fock space corresponds to the basis of Macdonald polynomials P λ . Wedenote by P the matrix with columns given by eigenvectors P λ . We denoteby E the diagonal matrix of eigenvalues, so that M (0) P = P E . The eigenvalues of E are monomials in t and t given by (38).From the basic theory of q -difference equations there exists a unique fun-damental solution of the QDE of the formΨ ( z ) = P Ψ reg ( z ) e ln( E
0) ln( z )ln( q ) , Ψ reg ( z ) = 1 + ∞ (cid:88) k =1 Ψ reg ,k z k (43)The matrix Ψ reg ( z ) solvesΨ reg ( zq ) E = M • ( z )Ψ reg ( z ) . (44)where M • ( z ) = P − M ( z ) P is the matrix for the operator M ( z ) in the basisof the fixed points P . Here, again, we assume implicitly that M ( z ) is the“matrix” of the corresponding operator in the basis p µ and P is the transitionmatrix from p µ to the basis of Macdonald polynomials P µ , see Remark 1above and Appendix B for examples of P .Let us define M • ( z ) := E − M • ( z ) , M • k ( z ) := E − k M • ( zq k ) E k . Then the infinite productΨ reg ( z ) = M • ( z ) − M • ( z ) − M • ( z ) − . . . (45)22olves (44). We denote by B • w ( z ) := P − B w ( z ) P , B • w := P − B w P the matrices of operators B w ( z ) and B w in the basis of fixed points P . Proposition 5.
For s ∈ Q the fundamental solution (45) has the followinglimits: lim q → Ψ reg ( zq s ) = , s ≥ (cid:16) ←− (cid:81) w ∈ ( s, ( B • w ) − (cid:17) · B • s ( z ) − s < Proof.
In the basis of fixed points P the matrix of the operator L is givenby E , thus M • ( z ) = −→ (cid:89) w ∈ [ − , B • w ( z )and from (39) we find M • k ( z ) = −→ (cid:89) w ∈ [ − − k, − k ) B • w ( z )which gives Ψ reg ( z ) = ←− (cid:89) w ∈ ( −∞ , B • w ( z ) − . The proposition follows from (37). z = ∞ By the general theory of quantum difference equations for Nakajima quivervarieties [28] the operator M ( z ) near z = ∞ defines the QDE for the Naka-jima variety with opposite stability condition. In particular, the matrix M ( ∞ ) is diagonalizable over Q ( t , t ) with eigenvalues given by monomialsin t , t .Let H ∗ be the matrix with columns given by the eigenvectors of M ( ∞ ).Let E ∞ be the diagonal matrix of eigenvalues so that: M ( ∞ ) H ∗ = H ∗ E ∞
23n Section 8.3 we will show that H ∗ = P ∗ with P ∗ as in Theorem 4, but atthe moment we only need the fact that the eigenbasis H ∗ exists. The QDEhas unique fundamental solution of the formΨ ∞ ( z ) = H ∗ Ψ reg ∞ ( z ) e ln( E ∞ ) ln( z )ln( q ) , Ψ reg ∞ ( z ) = 1 + ∞ (cid:88) k =1 Ψ reg ∞ ,k z − k . (46)The matrix Ψ reg ∞ ( z ) solves the equation:Ψ reg ∞ ( z ) E ∞ = M ∗ ( z )Ψ reg ∞ ( z ) (47)where M ∗ ( z ) = ( P ∗ ) − M ( z ) P ∗ denotes the matrix of M ( z ) in the basis ofeigenvectors P ∗ . We denote M ∗ ( z ) = E − ∞ M ∗ ( z ) , M ∗ k ( z ) = E k ∞ M ∗ ( zq − k ) E − k ∞ then the infinite productΨ reg ∞ ( z ) = M ∗ ( z ) M ∗ ( z ) M ∗ ( z ) · · · (48)solves (47). Let us denote B ∗ w ( z ) := ( P ∗ ) − B w ( z ) P ∗ , B ∗ w := ( P ∗ ) − B w P ∗ the matrices of the operators B w ( z ) and B w in the basis P ∗ . Proposition 6.
The solution (48) has the following limits: lim q → Ψ reg ∞ ( zq s ) = B ∗ s ( z − ) − ←− (cid:81) w ∈ [0 ,s ) ( B ∗ w ) − s ≥ , s < Proof.
Let us consider a finite approximation of infinite product (48):Ψ regN ( z ) = M ∗ ( z ) M ∗ ( z ) · · · M ∗ N ( z ) = M ∗ ( z/q ) M ∗ ( z/q ) · · · M ∗ ( z/q N ) E − N ∞ Using (39) we rewrite the last expression in the formΨ regN ( z ) = (cid:16) −→ (cid:89) [0 ,N ) B ∗ w ( z, q ) (cid:17) ( M (0) ∗ ) N E − N ∞ . q → Ψ regN ( zq s ) = B ∗ s ( z ) (cid:16) −→ (cid:89) ( s,N ) B ∗ w (cid:17) ( M (0) ∗ ) N E − N ∞ (49)Let us note that M ∗ ( z ) M ∗ ( z/q ) · · · M ∗ ( z/q N − ) = (cid:16) (cid:89) w ∈ [0 ,N ) B ∗ w ( z, q ) (cid:17) M ∗ (0) N where the last equality is by (39). At z = ∞ we obtain E N ∞ = (cid:16) (cid:89) w ∈ [0 ,N ) B ∗ w (cid:17) M ∗ (0) N which is the same as (cid:16) −→ (cid:89) [0 ,N ) B ∗ w (cid:17) ( M (0) ∗ ) N E − N ∞ = 1 . (50)Dividing (49) by (50) and assuming that N > s we obtainlim q → Ψ regN ( zq s ) = B ∗ s ( z ) ←− (cid:89) [0 ,s ] ( B ∗ w ) − = B ∗ s ( z − ) − ←− (cid:89) [0 ,s ) ( B ∗ w ) − where the last equality is by Proposition 4. We see that for large N , thelimit does not depend on N and the proposition follows. The monodromy of QDE is defined as the transition matrix between the twofundamental solutions constructed above:
Mon ( z, t , t ) := Ψ ( z ) − Ψ ∞ ( z ) . (51)Clearly, Mon ( zq, t , t ) = Mon ( z, t , t ). Remark 4.
If we change a basis of the Fock space by A ∈ End(
Fock ) thenthe fundamental solutions transforms asΨ ( z ) → A Ψ ( z ) , Ψ ∞ ( z ) → A Ψ ∞ ( z )Thus, Mon ( z, t , t ) does not depend on the basis in which the QDE isconsidered. 25t will be convenient to work with the regular part of monodromy which isdefined like (51) but without exponential factors in (43) and (46): Mon reg ( z, t , t ) := e ln( E
0) ln( z )ln( q ) Mon ( z, t , t ) e − ln( E ∞ ) ln( z )ln( q ) . (52)The regular part is not q -periodic Mon reg ( zq, t , t ) = E Mon reg ( z, t , t ) E − ∞ . (53)Combining Propositions 5 and 6 we obtain: Theorem 5.
For s ∈ Q the monodromy of qde has the following asymptoticat q → : lim q → Mon reg ( zq s , t , t ) = B • s ( z − ) − ←− (cid:81) w ∈ [0 ,s ) ( B • w ) − · T s ≥ , B • s ( z ) −→ (cid:81) w ∈ ( s, B • w · T , s < . (54) where B • w ( z ) , B • w denote the matrices of the operators B w ( z ) and B w in thebasis of fixed points P and T is the matrix of transition matrix between thebases P and H ∗ : T := P − H ∗ . (55) Remark 5.
The above theorem says that the limit lim q → Mon reg ( zq s , t , t ) isa piecewise constant function of s ∈ Q , which changes only when s crossesa “wall” from Walls n ⊂ Q . Moreover, if s (cid:54)∈ Walls n then B • s ( z ) = 1 and thelimit is independent on the K¨ahler variable z , see example below. Example 2.
Let us consider the case n = 3 and s = − / q → Mon reg ( zq − / , t , t ) = B •− / ( z ) B •− / B •− / T , but for n = 2 we obtainlim q → Mon reg ( zq − / ) = B •− / T , in particular, the last limit does not depend on z .26 Elliptic stable envelope
A new approach to studying the monodromy was recently suggested in [2, 24,25]. In this approach, the monodromy of the QDE for a variety X is identifiedwith the transition matrix between the elliptic stable bases in the equivariantelliptic cohomology of X . In this section we apply this idea to X = Hilb n ( C ).In this and next sections we will also use equivariant parameters a and (cid:126) defined by t = a (cid:126) / , t = a − (cid:126) / . Let us introduce the following functions:ˆ s ( x ) := x / − x − / , ϕ ( x ) := ∞ (cid:89) i =0 (1 − xq i )the last product converges for | q | < θ ( x ) := ϕ ( qx )ˆ s ( x ) ϕ ( qx − ) . (56)denote the q -theta function. We will denote by ˆ S , Φ, Θ multiplicative exten-sions of these functions to Laurent polynomials via the rules:ˆ S ( a + b ) = ˆ S ( a ) ˆ S ( b ) , Φ( a + b ) = Φ( a )Φ( b ) , Θ( a + b ) = Θ( a )Θ( b )Given a Laurent polynomial P we defile by det( P ) via det( a + b ) = ab. Example 3.
Let P = a − b + 3 c thenΦ( P ) = ϕ ( a ) ϕ ( c ) ϕ ( b ) , det( P ) = ac b . If P ∗ = a − − b − + 3 c − denote the dualization in K-theory thenΘ( P ) = Φ( qP ) ˆ S ( P )Φ( qP ∗ ) . The last formula, clearly, holds for any K-theory class P .27et V ∈ K T ( X ) be the class of the tautological bundle over the Hilbertscheme X . The class P = V + V ⊗ V ∗ t − V ⊗ V ∗ is a polarization of X , in other words, it is half of the class of the tangentbundle: T X = P + (cid:126) − P ∗ ∈ K T ( X ) . For the computations below, the following Lemma is convenient.
Lemma 2. s ( T X ) Θ( P )Φ(( q − (cid:126) ) P ) = Φ( qT X ) O (1) / (cid:126) − n/ t n / Proof.
Using the multiplicative notation we write:1ˆ s ( T X ) Θ( P )Φ(( q − (cid:126) ) P ) = 1ˆ s ( T X ) Φ( qP )ˆ s ( P )Φ( qP ∗ )Φ(( q − (cid:126) ) P ) = det( P ) / Φ( P ∗ + (cid:126) P )ˆ s ( T X ) . Since P ∗ + (cid:126) − P = T X we have1ˆ s ( T X ) Θ( P )Φ(( q − (cid:126) ) P ) = det( P ) / det( T X ) / Φ( qT X ) . Using the explicit form of the polarization and det( V ) = O (1) we computedet( P ) / = det( V ) / t n / = O (1) / t n / . As X is symplectic, with the T -weight of the symplectic form given by (cid:126) , foreach w ∈ weight T ( T λ X ) we have the symplectic dual dual weight w − (cid:126) − ∈ weight T ( T λ X ). Thus det( T X ) = (cid:126) − n . Lemma follows.
For a cocharacter σ ∈ Lie R ( A ) and a fixed point λ ∈ X T the construction of[2, 24] provides a class Stab Ellσ ( λ ) in the equivariant elliptic cohomology of28 , called the elliptic stable envelope of λ . In this paper we assume that σ corresponds to the cocharacter a → Ellσ ( λ ) can be characterized by their fixed point compo-nents U λ,µ ( a, z, (cid:126) ) := Stab Ellσ ( λ ) (cid:12)(cid:12) µ (57)By their definition, the components U λ,µ ( a, z, (cid:126) ) are sections of a certain linebundle over the abelian variety E = E z × E a × E (cid:126) where E = C × /q Z denotes the elliptic curve with modulus q . The parameters a, z, (cid:126) are viewedas coordinates on the factors. For the Hilbert scheme X the explicit formulasfor U λ,µ ( a, z, (cid:126) ) in terms of the theta functions were computed in [33].Let us denote by (cid:31) the standard dominance order on partitions. Fromthe support condition, in the definition of the elliptic stable envelope classes U λ,µ ( a, z, (cid:126) ) = 0 if λ (cid:31) µ . In other words, the matrix U λ,µ ( a, z, (cid:126) ) is uppertriangular if the fixed points are ordered by (cid:31) .The diagonal elements of this matrix are given explicitly by U λ,λ ( a, z, (cid:126) ) = Θ( N − λ ) := (cid:89) w ∈ char T ( Tλ X ) (cid:104) σ,w (cid:105) < θ ( w ) . where T λ X = N + λ ⊕ N − λ is the decomposition into repelling and attractingsubspaces for σ , and the product is over the T -weights of N + λ . Similarly, theelliptic stable envelopes for the opposite cocharacter are characterized by the lower triangular matrix L λ,µ ( a, z, (cid:126) ) := Stab Ell − σ ( λ ) (cid:12)(cid:12) µ (58)with L λ,µ ( a, z, (cid:126) ) = 0 if µ (cid:31) λ and L λ,λ ( a, z, (cid:126) ) = Θ( N + λ ) := (cid:89) w ∈ char T ( Tλ X ) (cid:104) σ,w (cid:105) > θ ( w ) . Let us consider the automorphism of the torus ι : T → T , ι ( t , t ) = ( t , t ) . ι ∗ be the corresponding induced maps in equivariant K-theory or el-liptic cohomology. On the classes of fixed points it acts by λ → λ (cid:48) ( λ (cid:48) denotesthe transposed partition). Note that ι ( a ) = a − and thus the correspondingmap of the Lie algebras maps the cocharacter σ to − σ . The uniqueness ofthe elliptic stable envelope classes thus implies that: ι ∗ (Stab Ellσ ( λ )) = Stab Ell − σ ( λ (cid:48) ) . In the fixed point components this gives:
Lemma 3.
The matrices of stable envelopes (57) and (58) are related by L ( a, z, (cid:126) ) = S U ( a − , z, (cid:126) ) S (59) where S is the antidiagonal matrix describing the transformation λ → λ (cid:48) . Inthe basis of partitions ordered by (cid:31) it has the form: S λ,µ := δ λ,µ (cid:48) = · · · · · · · · · · · · · · · . Comparing the diagonal elements of (59) we find S Θ( N − ) (cid:12)(cid:12) a =1 /a S = Θ( N + ) (60)The following property is also convenient for explicit computations: Lemma 4.
There are matrix identities L t ( a, z − (cid:126) − , (cid:126) ) U ( a, z, (cid:126) ) = Θ( T X ) where Θ( T X ) denotes the diagonal matrix with Θ( T X ) λ,λ = Θ( T λ X ) . Proof.
This identity is Proposition 3.4 in [2] applied to X .In this and following sections we use ˜ to denote the normalized matrices ofstable envelopes, defined by˜ U λ,µ ( a, z ) := U λ,µ ( a, z ) U µ,µ ( a, z ) , ˜ L λ,µ ( a, z ) := L λ,µ ( a, z ) L µ,µ ( a, z ) , so that ˜ U λ,λ ( a, z ) = ˜ L λ,λ ( a, z ) = 1. Note that the previous lemma impliesthat ˜ L t ( a, z − (cid:126) − ) ˜ U ( a, z ) = Id. (61)30 .4 Twisted elliptic stable envelopes
It is convenient to introduce twisted elliptic envelopes by
Stab
Ellσ ( λ ) = κ ∗ (Θ( N + λ )) Stab Ellσ ( λ ) (62)where κ ∗ denotes the change of variables κ ∗ : a → z √ (cid:126) , (cid:126) → / (cid:126) , z → a √ (cid:126) . (63)Note that the new prefactor κ ∗ (Θ( N ± + )) only depends on z and (cid:126) . We denotethe corresponding matrices of the fixed point components U λ,µ ( a, z ) := Stab
Ellσ ( λ ) (cid:12)(cid:12) µ , and as in Lemma 3 we define: L ( a, z ) := SU ( a − , z ) S Let κ ∗ (Θ( N ± )) be the diagonal matrices with diagonal κ ∗ (Θ( N ± λ )) then U ( a, z ) = κ ∗ (Θ( N + )) U ( a, z ) , L ( a, z ) = κ ∗ (Θ( N − )) (cid:12)(cid:12) z = z − (cid:126) − L ( a, z ) . (64) X The twisted version of the elliptic stable envelope (62) behaves better withrespect to the so called . Informally, the mirror con-jecture states that there exists a dual variety X ! so that the twisted ellipticstable envelopes of X and X ! coincide after identification of K¨ahler and equiv-ariant parameters by (63). In terms of the fixed point components this canbe formulated as follows. Conjecture 1.
The Hilbert scheme is self-dual with respect to D -mirrorsymmetry X ∼ = X ! and U ( a, z ) = κ ∗ (cid:16) L ( a, z − (cid:126) − ) t (cid:17) (65) where t denotes transposed matrix.
31y Lemma 4 this conjecture is equivalent to the identity:˜ U ( a, z ) − = κ ∗ ( ˜ U ( a, z ))For explicit examples of 3 D -mirror symmetry of the elliptic stable envelopewe refer to [30, 29, 34, 31], see also [5] for approach which uses vertex func-tions . The proof of 3 D -mirror symmetry for the Hilbert scheme (65) is awork in progress by several groups [16, 1]. Remark 6.
Conjecture 1 implies that the twisted stable envelopes are fixedpoint components of a certain class m in the equivariant elliptic cohomologyof X × X ! , called duality interface in [34]. Remark 7.
Conjecture 1 can be checked directly for several first values of n using a computer and the explicit formulas from [33]. The one-leg vertex functions with a descendent τ ∈ K T ( X ): V ( τ )0 ( z ) ∈ K T ( X )[[ z ]]of the Hilbert scheme X are defined as the generating series of quasimaps to X , see Section 7.2 of [23] for the definition. The vertex functions are certaingeneralizations of hypergeometric functions. In particular, they provide abasis to a certain system of q -hypergeometric equations, which are holomor-phic near z = 0 and satisfy V ( τ )0 (0) = τ . Solving the same hypergeometricsystem at z = ∞ with the same boundary conditions gives functions V ( τ ) ∞ ( z ) ∈ K T ( X )[[ z − ]]The functions V ( τ ) ∞ ( z ) can be considered as vertex functions for the Nakajimavariety associated with the same combinatorial data (quiver and dimensionvectors) but with the opposite stability condition. We need the followingresult of A.Okounkov [25], which describes the monodromy of the properlynormalized vertex functions. Theorem 6.
Let us consider the vertex functions of X normalized so that ˜ V ( z ) = Φ(( q − (cid:126) ) P )Θ( P ) V ( z ) , ˜ V ∞ ( z ) = Φ(( q − (cid:126) ) P )Θ( P ) V ∞ ( z ) (66)32 hen ˜ V ∞ ( z ) = Λ ( z ) ˜ V ( z ) (67) where the monodromy matrix is expressed via the elliptic stable envelopes (64)by: Λ ( z ) = ( − n L ( a − , z − (cid:126) − ) − U ( a, z ) . (68) Proof.
This is Corollary 3.2 in [25] applied to Hilbert scheme X = Hilb n ( C ).We only need to explain the sign ( − n : in [25] the normalization of stableenvelope is fixed by formula (45) in [2]. It differs from the one accepted inthis paper by ( − rk(ind σλ ) where ind σλ denotes the index of the fixed point λ corresponding to the chamber σ . Thus, the ratio of stable envelopes L ( a, (cid:126) )and U ( a, z ) differs from those in Corollary 3.2 of [25] by a sign( − rk(ind + σλ ) − rk(ind − σλ ) = ( − n The last identity is by direct computation as in Section 3.8 of [33].The fundamental solutions Ψ ( z ), Ψ ∞ ( z ) play the role of the cappingoperators in the enumerative geometry. The relation between the bare vertexfunctions, capping operators and capped vertex functions is the following, seeSection 7.4 of [23]:Ψ ( z ) ˆ s ( T X ) − V ( τ )0 ( z ) = Ψ ∞ ( z ) ˆ s ( T X ) − V ( τ ) ∞ ( z ) = (cid:104) τ (cid:105) ∈ K T ( X )( z ) (69)where (cid:104) τ (cid:105) ∈ K T ( X )( z ) is known as the capped vertex with a descendent τ .As a rational function of z , the capped vertex (cid:104) τ (cid:105) has trivial monodromy.Thus, the equation (69) says that the monodromy of the vertex functionand the capping operators (the solutions of QDE) are inverses of each other.Combining all the factors together we find: Theorem 7.
Let us normalize solutions of QDE by ˜Ψ ( z ) = Ψ ( z )Φ( q T X ) , ˜Ψ ∞ ( z ) = Ψ ∞ ( z )Φ( q T X ) (70) where Φ( q T X ) denotes the diagonal operator Φ( q T X ) λ,λ = (cid:89) w ∈{ T − weights of T λ X } ϕ ( qw )33 the product runs over the T -weights appearing in the tangent space T λ X ata fixed point λ ∈ X T ). Then, we have ˜Ψ ( z ) = ˜Ψ ∞ ( z ) ˜ Λ ( z ) where ˜ Λ ( z ) = O (1) / Λ ( z ) O (1) − / . and Λ ( z ) is given by (68) . Proof.
Applying Lemma 2 to normalizations (66) and (70) we write (69) inthe form: ˜Ψ ( z ) O (1) / ˜ V ( τ )0 ( z ) = ˜Ψ ∞ ( z ) O (1) / ˜ V ( τ ) ∞ ( z )The theorem then follows from (67).For the monodromy (51) we obtain Corollary 1.
The monodromy of the QDE equals:
Mon reg ( z ) =( − n Φ( q T X ) O (1) / U ( a, z ) − L ( a − , z − (cid:126) − ) O (1) − / Φ( q T X ) − (71) Remark 8.
Note that, in this corollary,Φ( q T X ) O (1) / U ( a, z ) − is an upper triangular and L ( a − , z − (cid:126) − ) O (1) / Φ( q T X ) − is a lower-triangular matrix. Thus, the corollary provides a Gauss decompo-sition of the monodromy.
Theorem 5 says that the wall-crossing operators B • w ( z ) appear as q → Mon reg ( z ). Thus, by Corollary 1 these operatorscan be expressed via limits of elliptic stable envelopes U ( a, z ) and L ( a, z ). Inthis section, following [14, 15] we show that q → U ( a, z )factors into product of K-theoretic stable envelopes of X and its 3D-mirror X ! .As a result, we obtain natural Gauss decomposition of matrices the B • w ( z )and B • w expressed via K-theoretic stable classes of X and X ! .34 .1 K-theoretic stable envelope In the limit q →
0, the elliptic cohomology scheme of X degenerates to thescheme spec( K T ( X )). The limit of the elliptic stable envelope then gives asection of the trivial line bundle over Spec( K T ( X )), i.e., the K-theory class.Here is the precise statement: Theorem 8.
For generic s ∈ H ( X , Q ) ∼ = Q we have lim q → Stab
Ellσ ( λ ) (cid:12)(cid:12) z = zq s = Stab X ,Kth, [ s ] σ ( λ ) ∈ K T ( X ) ⊗ Q [ t ± / , t ± / ] (72) where Stab X ,Kth, [ s ] σ ( λ ) is the K-theoretic stable envelope of a fixed point λ witha slope s . In particular, for the matrices of the fixed point components (57)we have lim q → U λ,µ ( a, zq s ) = Stab X ,Kth, [ s ] σ ( λ ) (cid:12)(cid:12)(cid:12) µ ∈ Q [ t ± / , t ± / ] . For the definition of K-theory classes Stab X ,Kth, [ s ] σ ( λ ) we refer to Section9 of [23] or Section 2 of [28]. Proof.
Proposition 4.3 in [2].
Remark 9.
We note that in [2], to get rid of square roots, the K-theoreticlimit is additionally twisted by the square root of the polarization. In thiscase, the limit is an element of K -theory:det( P / ) ⊗ lim q → Stab
Ellσ ( λ ) (cid:12)(cid:12) z = zq s ∈ K T ( X )We will use (72) as definition of K-theoretic stable envelope in this paper. Itdiffers from the K-theoretic stable envelope of [2] by the factor det( P / ). For the Hilbert scheme X , by generic slope in Theorem 8 we mean s ∈ Q \ Walls n . For this paper we need a more general version of Theorem 8,which includes the limits for non-generic slopes s ∈ Walls n . The limits of thiskind were studied in [15], in particular see Section 8 in [15] for discussion ofthe Hilbert scheme X . 35or w = ab ∈ Q we consider the following cyclic subgroup of A : µ w = { e πikw : k = 0 , . . . , b − } ⊂ A (73)As a subgroup of A is acts naturally on X . Let Y w = X µ w ⊂ X . be its fixed point set. Proposition 7.
The subvariety Y w is a union of connected components: Y w = (cid:97) n + ··· + n b − = n X ( n , . . . , n b − ) (74) where X ( n , . . . , n b − ) is isomorphic to the Nakajima variety associated withcyclic quiver of length b with dimension vector v = ( n , . . . , n b − ) and framingvector w = (1 , , . . . , , see Fig 2: Figure 2: A cyclic quiver with dimension vector v and framing vector w . Proof.
Proposition 6 in [14].
Remark 10.
We note that if at least one n i = 0 then the cyclic quivervariety X ( n , . . . , n b − ) ∼ = pt or empty, see [6] for combinatorial description of zero dimensional A n -typequiver varieties. In particular for w = ab with b > n all components of Y w = X T are points. This can be summarized as: { w ∈ Q : Y w (cid:54) = Y T } = Walls n . (75)36 .3 K-theory limit of the elliptic stable envelopes Let U + w ( a ) (respectively U − w ( a )) denote the matrix of fixed point componentsof the K-theoretic stable envelopes of X with slope w + (cid:15) (respectively w − (cid:15) )for small ample 0 < (cid:15) (cid:28) U ± w ( a ) λ,µ := Stab X ,Kth,w ± (cid:15)σ ( λ ) (cid:12)(cid:12) µ , L ± w ( a ) λ,µ := Stab X ,Kth,w ± (cid:15) − σ ( λ ) (cid:12)(cid:12)(cid:12) µ . (76)We will also need the normalized matrices:˜ U ± w ( a ) λ,µ := U ± w ( a ) λ,µ U ± w ( a ) µ,µ , ˜ L ± w ( a ) λ,µ := L ± w ( a ) λ,µ L ± w ( a ) µ,µ . Matrices (76) in cases n = 2 , Remark 11. If w (cid:54)∈ Walls n then U + w ( a ) = U − w ( a ). It is also known that U + w ( a ) = U − w ( a ), L + w ( a ) = L − w ( a ) for w ∈ Z , see [12].Similarly we denote by U + Y w , L + Y w (respectively U − Y w , L − Y w ) the matricesof K -theoretic stable envelopes for the cyclic quiver variety variety Y w withsmall ample (respectively anti-ample) slope: U ± Y w ( a ) λ,µ = Stab Y w ,Kth, ± (cid:15)σ ( λ ) (cid:12)(cid:12) µ , L ± Y w ( a ) λ,µ = Stab Y w ,Kth, ± (cid:15) − σ ( λ ) (cid:12)(cid:12)(cid:12) µ (77)We denote the corresponding normalized matrices by˜ U ± Y w ( a ) λ,µ = U ± Y w ( a ) λ,µ U ± Y w ( a ) µ,µ , ˜ L ± Y w ( a ) λ,µ = L ± Y w ( a ) λ,µ L ± Y w ( a ) µ,µ . Matrices (77) for n = 2 , Lemma 5.
There are matrix identities: (cid:16) ˜ L ± Y w ( a ) (cid:17) t ˜ U ∓ Y w ( a ) = 1 , (cid:16) ˜ L ± w ( a ) (cid:17) t ˜ U ∓− w ( a ) = 1 . Applying Theorem 8 to Lemma 3 we obtain:
Lemma 6.
There are matrix identities: L ± Y w ( a ) = S U ± Y w ( a − ) S , L ± w ( a ) = S U ± w ( a − ) S .
37e are ready to formulate the main theorem of this section, describing theK-theoretic limit of the elliptic stable envelope for arbitrary slopes:
Theorem 9.
If the mirror symmetry Conjecture 1 holds, then for all w ∈ H ( X , Q ) ∼ = Q we have the following limits for the elliptic stable envelopematrix (57): lim q → U ( a, zq w ) = γ w ( a ) κ ∗ ( ˜ L ∓ Y w ( a )) t γ w ( a ) − U ± w ( a ) Here κ ∗ denotes substitution (63) and γ w ( a ) denotes the diagonal matrix withelements γ w ( a ) λ,λ = (cid:89) (cid:3) ∈ λ ( − a ) wc λ ( (cid:3) ) ( − (cid:126) − ) (cid:98) wh λ ( (cid:3) ) (cid:99) + . (78) where c λ ( (cid:3) ) and h λ ( (cid:3) ) are the standard content and hook lengths of a box (cid:3) in a Young diagram λ .Proof. Theorem 3 and Theorem 4 in [15] applied to the Hilbert scheme X . Remark 12.
For generic w (cid:54)∈ Walls n by (75) we have ˜ L ∓ Y w ( a ) = 1 and the lasttheorem reduces to the Aganagic-Okounkov K-theoretic limit - Theorem 8. Remark 13.
By Lemma 5 we have( ˜ U ± Y w ( a )) − = ( ˜ L ∓ Y w ( a )) t and the above theorem can be also written aslim q → U ( a, zq w ) = γ w ( a ) κ ∗ ( ˜ U ± Y w ( a )) − γ w ( a ) − U ± w ( a ) . (79)The following identity will be convenient for computations: Lemma 7.
In the basis of fixed points the operator γ w ( a ) − S γ − w ( a ) S acts by γ w ( a ) − S γ − w ( a ) S : P λ → (cid:16) ( − codim X ( Y w )2 (cid:126) codim X ( Y w )4 (cid:89) (cid:3) ∈ λ (cid:126) (cid:98) h λ ( (cid:3) ) · w (cid:99) (cid:17) P λ where codim X ( Y w ) denotes the codimension in X of the component of (74)containing the fixed point λ . .4 Gauss decomposition of B • w ( z ) Theorem 9 suggests to introduce the following matrices: U ± Y w ( a, z, (cid:126) ) := γ w ( a ) κ ∗ ( U ± Y w ( a )) , L ± Y w ( a, z, (cid:126) ) := S γ − w ( a ) κ ∗ ( U ± Y w ( a − )) S . (80)By Lemma 6 the last one takes the form L ± Y w ( a, z, (cid:126) ) := S γ − w ( a ) S κ ∗ ( L ± Y w ( a )) . We are ready to formulate the main result of this section:
Theorem 10.
For all w ∈ Q and l , l ∈ { + , −} we have the followingmatrix identitiesif w < then (cid:16) B • w ( z ) −→ (cid:81) s ∈ ( w, B • s (cid:17) · T =( − n U l w ( a, (cid:126) ) − U l Y w ( a, z, (cid:126) ) L l Y w ( a, z, (cid:126) ) − L l − w ( a − , (cid:126) ) , if w ≥ then (cid:16) B • w ( z − ) − ←− (cid:81) s ∈ [0 ,w ) ( B • s ) − (cid:17) · T =( − n U l w ( a, (cid:126) ) − U l Y w ( a, z, (cid:126) ) L l Y w ( a, z, (cid:126) ) − L l − w ( a − , (cid:126) ) where B • w ( z ) and B • w denote the matrices of operators B w ( z ) and B w in thebasis of fixed points. Remark 14.
The matrices U ± w ( a, (cid:126) ) , U ± Y w ( a, z, (cid:126) ) are upper-triangular and L ± w ( a ) , L ± Y w ( a, z, (cid:126) ) are lower-triangular if the fixed point basis is orderedby (cid:31) , see examples in Appendix C. Thus, the above theorem describes aGauss decomposition of the matrices B • w ( z ), B • w . Proof.
We compute the limit of monodromy (54) using Gauss decomposi-tion (71). First we note that the factors Φ( q T X ) in (71) do not depend on z and lim q → Φ( q T X ) = 1 . N ± ) the diagonal matrix with diagonal elementsΘ( N + λ ) = (cid:89) (cid:3) ∈ λ ϑ ( a h λ ( (cid:3) ) (cid:126) / − a λ ( (cid:3) )+ l λ ( (cid:3) )+1) ) , Θ( N − λ ) = (cid:89) (cid:3) ∈ λ ϑ ( a − h λ ( (cid:3) ) (cid:126) / a λ ( (cid:3) ) − l λ ( (cid:3) )+1) ) . By (60) we have S Θ( N − λ ) S = Θ( N + λ ) (cid:12)(cid:12) a = a − . Using this identity and (64) wewrite (71) as a product of three factors: U ( z, a ) − κ ∗ (cid:16) Θ( N − λ )Θ( N + λ ) (cid:17) S U ( z − (cid:126) − , a ) S . We compute the limits of these factors separately. First, by (79) we havelim q → U ( zq w , a ) − = U l w ( a ) − γ w ( a ) κ ∗ ( ˜ U l Y w ( a )) γ w ( a ) − , for any l ∈ { + , −} and by (80) we obtainlim q → U ( zq w , a ) − = U l w ( a ) − U l Y w ( a, z, (cid:126) ) γ w ( a ) − . Similarly, we computelim q → S U ( z − (cid:126) − q − w , a ) S = S γ − w ( a ) κ ∗ ( ˜ U l Y w ( a − ) − ) γ − w ( a ) − U l − w ( a ) S , for any l ∈ { + , −} . As S = 1 we can write the last expression in the form S γ − w ( a ) S S κ ∗ ( ˜ U l Y w ( a − ) − ) γ − w ( a ) − S S U l − w ( a ) S , By Lemma 6 we have
S U l − w ( a ) S = L l − w ( a − ) and by (80) we obtain:lim q → S U ( z − (cid:126) − q − w , a ) S = S γ − w ( a ) S L l Y w ( a, z, (cid:126) ) − L l − w ( a − ) . Let us denote by N ± λ ( Y w ) the attracting and repelling parts of T λ Y w . Bydefinition, these are the subspaces N ± λ ( Y w ) ⊂ N ± λ invariant with respect tothe action of cyclic group (73) , thusΘ( N + λ ( Y w )) = (cid:89) (cid:3) ∈ λ,hλ ( (cid:3) ) w ∈ Z ϑ ( a h λ ( (cid:3) ) (cid:126) / − a λ ( (cid:3) )+ l λ ( (cid:3) )+1) ) ,
40r Θ( N − λ ( Y w )) = (cid:89) (cid:3) ∈ λ,hλ ( (cid:3) ) w ∈ Z ϑ ( a − h λ ( (cid:3) ) (cid:126) / a λ ( (cid:3) ) − l λ ( (cid:3) )+1) ) . Computing the limits of the theta functions using (8) in [14], we find:lim q → κ ∗ (cid:16) Θ( N − λ )Θ( N + λ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) z = zq w = κ ∗ (cid:16) ˆ s ( N − λ ( Y w ))ˆ s ( N + λ ( Y w )) (cid:17) ( − codim X ( Y w )2 (cid:126) − codim X ( Y w )4 (cid:89) (cid:3) ∈ λ (cid:126) −(cid:98) w · h λ ( (cid:3) ) (cid:99) Combining all these factors together and using Lemma 7: γ w ( a ) − S γ − w ( a ) S = ( − codim X ( Y w )2 (cid:126) codim X ( Y w )4 (cid:89) (cid:3) ∈ λ (cid:126) (cid:98) w · h λ ( (cid:3) ) (cid:99) . we arrive at the statement of the theorem. • w Theorem 11.
For all w ∈ Q \ Walls n the matrices of monodromy operatorsin the basis of fixed points have the following Gauss decomposition: ( − n U + w ( a, (cid:126) ) − D w ( (cid:126) ) L −− w ( a − , (cid:126) ) = −→ (cid:81) s ∈ ( w, B • s · T , w < ←− (cid:81) s ∈ [0 ,w ] ( B • s ) − · T , w ≥ where D w ( (cid:126) ) is the diagonal matrix with eigenvalues D w ( (cid:126) ) λ,λ = ( − n (cid:126) − n/ (cid:89) (cid:3) ∈ λ (cid:126) −(cid:98) h λ ( (cid:3) ) · w (cid:99) . (81) Proof.
Assume w <
0. Let w (cid:48) = w + (cid:15) be a regular slope, i.e., w (cid:48) is not awall. We apply Theorem 10 to w (cid:48) which gives B • w (cid:48) ( z ) −→ (cid:89) s ∈ ( w (cid:48) , B • s T = ( − n U l w (cid:48) ( a, (cid:126) ) − U l Y w (cid:48) ( a, z, (cid:126) ) L l Y w (cid:48) ( a, z, (cid:126) ) − L l − w (cid:48) ( a − , (cid:126) )As w (cid:48) is not a wall B • w (cid:48) ( z ) = 1 and U ± Y w (cid:48) ( a, z, (cid:126) ) = γ w (cid:48) ( a ) , L ± Y w (cid:48) ( a, z, (cid:126) ) = S γ − w (cid:48) ( a ) S . γ w (cid:48) ( a ) S γ − w (cid:48) ( a ) − S = ( − codim X ( Y w (cid:48) )2 (cid:126) − codim X ( Y w (cid:48) )4 (cid:89) (cid:3) ∈ λ (cid:126) −(cid:98) h λ ( (cid:3) ) · w (cid:48) (cid:99) . Again, as w (cid:48) is not a wall, Y w (cid:48) is zero-dimensional, see Remark 10. Thuscodim X ( Y w (cid:48) ) = dim( X ) = 2 n. Finally, by our choice of w (cid:48) we have: U ± w (cid:48) ( a, (cid:126) ) = U + w ( a, (cid:126) ) , L ±− w (cid:48) ( a, (cid:126) ) = L −− w ( a, (cid:126) ) , (cid:98) h λ ( (cid:3) ) · w (cid:48) (cid:99) = (cid:98) h λ ( (cid:3) ) · w (cid:99) . Combining all these factors together, we arrive at the statement of the the-orem. For w >
The goal of this section is to relate the wall-crossing operators B w ( z ) andthe monodromy operators B w to the K-theoretic R -matrices of the Hilbertscheme X and the cyclic quiver varieties Y w ⊂ X ! . In the next subsection webriefly recall the definition of the K-theoretic R -matrices and review theirbasic properties. For more more systematic introduction we refer to Section2 of [28], Section 9 of [23] or Negut’s thesis [22]. R -matrices We recall that for generic choices of a cocharacter σ ∈ cochar( A ) and a slope s ∈ H ( X , Q ) the K-theoretic stable envelopes Stab X ,Kth, [ s ] σ ( λ ) of torus fixedpoints λ ∈ X T provide bases of the localized K-theory of X . The stableenvelopes only change when σ crosses certain hyperplanes in Lie Q ( A ) or s crosses certain hyperplanes in H ( X , Q ). The K-theoretic R -matrices aredefined as the corresponding transition matrices between the stable bases. Definition 1.
The total K -theoretic R - matrix of a variety X with slope s ∈ H ( X, Q ) is the transition matrix from the stable basis Stab
X,Kth, [ s ] − σ tothe stable basis Stab
X,Kth, [ s ] σ R ± Y w denote the total R -matrix of the cyclic quiver variety Y w withsmall ample or anti-ample slopes ± (cid:15) ∈ H ( Y w , Q ). Using our notations (77)we find R ± Y w ( a ) = U ± Y w ( a ) L ± Y w ( a ) − . (82)Note that this formula provides a Gauss decomposition of R ± Y w ( a ). Definition 2.
The wall R - matrix of a variety X is the transition matrixfrom the stable basis Stab
X,Kth, [ s ] σ to the stable basis Stab
X,Kth, [ s (cid:48) ] σ where s and s (cid:48) are two slopes separated by a single wall w such that s (cid:48) − s is ample. For instance, the wall R -matrices of the Hilbert scheme X has the form: R X , + σwall, w = U − w ( a ) U + w ( a ) − , R X , − σwall, w = L − w ( a ) L + w ( a ) − . We will also need the twisted version of K-theoretic R -matrix of Y w : R ± Y w ( z ) = γ w ( a ) κ ∗ ( R ± Y w ( a )) γ w ( a ) − (83)Explicit examples of R ± Y w ( a ) and R ± Y w ( z ) for n = 2 and n = 3 can be foundin Appendix E. Here we list main properties of R ± Y w ( z ): Proposition 8.
For two partitions λ, µ and w ∈ Q let d λ,µ ( w ) = w ( (cid:88) (cid:3) ∈ λ c λ ( (cid:3) ) − (cid:88) (cid:3) ∈ µ c µ ( (cid:3) )) then the matrix coefficients of R ± Y w ( z ) are • R ± Y w ( z ) λ,µ ∈ Q ( z, (cid:126) , a ) , • if d λ,µ ( w ) (cid:54)∈ Z then R ± Y w ( z ) λ,µ = 0 , • if d λ,µ ( w ) ∈ Z then R ± Y w ( z ) λ,µ = a d λ,µ ( w ) c λ,µ for c λ,µ ∈ Q ( z, (cid:126) ) .Proof. It is obvious from the definition of γ w ( a ) that R ± Y w ( z ) λ,µ ∼ a d λ,µ ( w ) The vanishing of matrix elements for non-integral d λ,µ ( w ) follows immediatelyfrom Theorem 3 of [15]. 43ere is another elementary property of the total K-theoretic R -matrices: Proposition 9.
Assume X is a symplectic variety for which the K -theoreticstable envelope exist, then the total K -theoretic R-matrices of X have thefollowing unitary property R [ s ] X ( a ) − = R [ − s ] X ( a − ) , in particular for small slopes s = ± (cid:15) we have R ± X ( a ) − = R ∓ X ( a − ) . Proof.
The proof follows the logic of Sections 4.5.1 - 4.5.3 in [17].For R -matrices R ± Y w ( a ) this proposition gives (110). For n = 2 , R -matrices of X : Proposition 10.
The matrix elements of the wall R-matrices for the Hilbertscheme X have the following properties: • ( R X , ± σwall, w ) λ,µ ∈ Q [ a ± , (cid:126) ± ] , • ( R X , + σwall, w ) λ,µ = ( R X , − σwall, w ) µ,λ = 0 if λ (cid:31) µ , i.e. the wall R -matrices areupper and lower triangular of the fixed points ordered by (cid:31) . • ( R X , ± σwall, w ) λ,λ = 1 , • if d λ,µ ( w ) (cid:54)∈ Z then ( R X , ± σwall, w ) λ,µ = 0 , • if d λ,µ ( w ) ∈ Z then ( R X , ± σwall, w ) λ,µ ∼ a ± d λ,µ ( w ) .Proof. All properties follow immediately from Theorem 1 of [28]. w ( z ) as R -matrices of Y w Let (cid:121) B w ( z ) = { (cid:121) B w ( z ) λ,µ : λ, µ ∈ X T } denote the matrix of the operator B w ( z )in the mixed stable basis : the input is the stable basis before a wall w , s w − (cid:15)λ := Stab X ,Kth, [ w − (cid:15) ] σ ( λ ) (84)44nd the output in the stable basis after w : s w + (cid:15)λ := Stab X ,Kth, [ w + (cid:15) ] σ ( λ ) . (85)Explicitly, we have B w ( z )( s w − (cid:15)λ ) = (cid:88) µ,λ (cid:121) B w ( z ) λ,µ s w + (cid:15)λ . (86)As usual, 0 < (cid:15) (cid:28) (cid:121) B w := (cid:121) B w ( ∞ ), the matrix of the monodromy operators B w in these bases. Theorem 12.
The matrices (cid:121) B w ( z ) coincide with the K -theoretic R -matricesof Y w : (cid:121) B w ( z ) = (cid:126) Ω w R − Y w ( z ) , (cid:121) B w = (cid:126) Ω w R − Y w ( ∞ ) , where (cid:126) Ω w := ( − (cid:126) / ) n − codim X ( Y w )2 . (87) Proof.
Assume w < l = l = − we have B • w ( z ) −→ (cid:89) s ∈ ( w, B • s T = ( − n U − w ( a, (cid:126) ) − U − Y w ( a, z, (cid:126) ) L − Y w ( a, z, (cid:126) ) − L −− w ( a − , (cid:126) )(88)By Theorem 11 we have −→ (cid:89) s ∈ ( w, B • s T = ( − n U + w ( a, (cid:126) ) − D w ( a, (cid:126) ) L −− w ( a − , (cid:126) ) (89)All operators here are invertible, thus dividing (88) by (89) from the rightwe obtain: B • w ( z ) = U − w ( a, (cid:126) ) − U − Y w ( a, z, (cid:126) ) L − Y w ( a, z, (cid:126) ) − D w ( a, (cid:126) ) − U + w ( a, (cid:126) ) (90)We have: U − Y w ( a, z, (cid:126) ) L − Y w ( a, z, (cid:126) ) − = γ w ( a ) κ ∗ ( U − Y w ( a ) L − Y w ( a ) − ) S γ − w ( a ) − S S γ − w ( a ) − S = γ w ( a ) − ( − (cid:126) / ) − codim X ( Y w )2 (cid:89) (cid:3) ∈ λ (cid:126) −(cid:98) h λ ( (cid:3) ) · w (cid:99) which together with (81) gives B • w ( z ) = U − w ( a, (cid:126) ) − (cid:126) Ω R − Y w ( z ) U + w ( a, (cid:126) )or, equivalently (cid:126) Ω R − Y w ( z ) = U − w ( a, (cid:126) ) B • w ( z ) U + w ( a, (cid:126) ) − . By definition, B • w ( z ) is the matrix of the operator B w ( z ) in the basis of fixedpoints and U ± w ( a, (cid:126) ) are the transition matrices from the basis of torus fixedpoints to stable bases s w ± (cid:15)λ . Thus, the last equality means that (cid:121) B w ( z ) = (cid:126) Ω R − Y w ( z ) . (91)For w ≥ l = l = − we obtain: (cid:16) B • w ( z − ) − ←− (cid:89) s ∈ [0 ,w ) ( B • s ) − (cid:17) · T = U − w ( a, (cid:126) ) − U − Y w ( a, z, (cid:126) ) L − Y w ( a, z, (cid:126) ) − L −− w ( a − , (cid:126) )By Theorem 11 we have ←− (cid:89) s ∈ [0 ,w ] ( B • s ) − · T = U + w ( a, (cid:126) ) − D w ( a, (cid:126) ) L −− w ( a − , (cid:126) )dividing first by the second we obtain B • w ( z − ) − B • w = U − w ( a, (cid:126) ) − U − Y w ( a, z, (cid:126) ) L − Y w ( a, z, (cid:126) ) − D w ( a, (cid:126) ) − U + w ( a, (cid:126) )Finally, Proposition 4 gives B • w ( z − ) − B • w = B • w ( z ) and we arrive at the sameidentity (90) as in w < z = ∞ in (91) we obtain (cid:121) B w = R − Y w ( ∞ ) . Remark 15.
Instead of (84)-(85) we could consider the stable bases foropposite chamber − σ : s w ± (cid:15)λ = Stab X ,Kth, [ w ± (cid:15) ] − σ and define the matrices (cid:121) B w ( z ) by (86). Then Theorem 12 would take theform (cid:121) B w ( z ) = (cid:126) Ω w R + Y w ( z ) , (cid:121) B w = (cid:126) Ω w R + Y w ( ∞ ) . .3 Monodromy operators B w as wall R -matrices Theorem 13.
There are following matrix identities: R − Y w (0) = (cid:126) − Ω w R X , + σwall, w , R + Y w (0) = (cid:126) − Ω w R X , − σwall, w . (92) Proof.
By Theorem 12 we have (cid:121) B w ( z ) = (cid:126) Ω w R − Y w ( z )which means (cid:126) Ω w R − Y w ( z ) = U − w ( a, (cid:126) ) B • w ( z ) U + w ( a, (cid:126) ) − Evaluating this at z = 0 and noting that B • w (0) = 1 we obtain: R − Y w ( z ) = (cid:126) − Ω w U − w ( a, (cid:126) ) U + w ( a, (cid:126) ) − = (cid:126) − Ω w R X , + σwall, w . Changing σ → − σ we obtain (cid:121) B w ( z ) = (cid:126) Ω w R + Y w ( z )see Remark 15 above. At z = 0 this gives the second identity. Remark 16.
Identities (92) are equivalent to Theorem 8 in [15]. They relatethe wall R -matrices of X with slopes ± (cid:15) to the R -matrices of Y w .Let us consider w = pm ∈ Q , with gcd ( p, m ) = 1. We note that the variety Y w is defined as the fixed subset of the finite group (73). In particular, itonly depends on the denominator m of w . Thus, the operators (cid:126) Ω w and R ± Y w ( a ) depend only on m as well. The Theorem 13 and definition of twisted R -matrix (83) then provides the following result about the wall R-matricesfor the Hilbert scheme X : Corollary 2.
Let w = pm , w (cid:48) = p (cid:48) m , with gcd ( p, m ) = gcd ( p (cid:48) , m ) = 1 then γ w ( a ) − R X , ± σwall, w γ w ( a ) = γ w (cid:48) ( a ) − R X , ± σwall, w (cid:48) γ w (cid:48) ( a ) (93) and the matrix elements of (93) are elements of Q [ (cid:126) ± ] . .4 QDE as a non-minuscule qKZ equation Let s − (cid:15)λ be the stable basis of K T ( X ) with small anti-ample slope − (cid:15) , i.e., thebasis (84) for w = 0. Let E be the diagonal matrix with eigenvalues givenby O X (1) | λ ∈ K T ( pt ), as defined in Section 4. Theorem 14.
In the stable basis s − (cid:15)λ the matrix of the operator M L ( z ) (defined by (41)) takes the form E −→ (cid:89) w ∈ [ − , (cid:126) Ω w R − Y w ( zq − w ) . (94) Proof.
Theorem 12 gives (cid:126) Ω w R − Y w ( z ) = U − w ( a, (cid:126) ) B • w ( z ) U + w ( a, (cid:126) ) − . By (35) we have B • w ( zq − w ) = B • w ( z, q ) and thus (cid:126) Ω w R − Y w ( zq − w ) = U − w ( a, (cid:126) ) B • w ( z, q ) U + w ( a, (cid:126) ) − . Assume the product in (94) is over the walls w , w , . . . , w m with w i < w i +1 .By definition U + w i ( a, (cid:126) ) = U − w i +1 ( a, (cid:126) ) , U + w m ( a, (cid:126) ) = U − ( a, (cid:126) )Thus we compute E −→ (cid:89) w ∈ [ − , (cid:126) Ω w R − Y w ( zq − w ) = E U −− ( a, (cid:126) ) (cid:16) −→ (cid:89) w ∈ [ − , B • w ( z, q ) (cid:17) U − ( a, (cid:126) ) − Twists by line bundles change the slope of the stable bases by integral shifts,in particular E U −− ( a ) − = U − ( a ) − E We obtain: E −→ (cid:89) w ∈ [ − , (cid:126) Ω w R − Y w ( zq − w ) = U − ( a, (cid:126) ) M • L ( z ) U − ( a, (cid:126) ) − where M • L ( z ) = E (cid:16) −→ (cid:89) w ∈ [ − , B • w ( z, q ) (cid:17) M L ( z ) in the basis of fixed points. Bydefinition, U − ( a ) is the transition matrix between the stable basis s − (cid:15)λ andthe basis of fixed points. The theorem follows.By above theorem, the quantum difference equation (40) for the Hilbertscheme X , in the stable basis s − (cid:15)λ takes the formΨ( zq ) = E (cid:16) −→ (cid:89) w ∈ [ − , (cid:126) Ω w R − Y w ( zq − w ) (cid:17) Ψ( z ) (95)where E is a diagonal matrix with eigenvalues given by monomials in a and (cid:126) . The q -difference equations of this type, which include the productsof “trigonometric” (i.e. K-theoretic) R-matrices shifted by powers of q ap-pear in mathematical physics as the quantum Knizhnik-Zamolodchikov (qKZ)equations [11]. The equation (95) is the proper version of the qKZ equationfor the toroidal algebra U (cid:126) ( (cid:98)(cid:98) gl ) associated with the Hilbert scheme X .The difference between (95) and the standard qKZ equations is that R − Y w ( z ) for different walls w are allowed to be the trigonometric R -matricesof different quantum groups . Indeed, the Nakajima varieties Y w for differentwalls w = p/N ∈ [ − ,
0) corresponds to cyclic quivers which may have dif-ferent length N ≤ n . The matrices R − Y w ( z ) appearing in (95) correspond tothe trigonometric R -matrices of the quantum toroidal algebras U (cid:126) ( (cid:98)(cid:98) gl N ) with N varying in the interval 1 < N ≤ n . Let ψ ( z ), ψ ∞ ( z ) be the fundamental solution matrices of the qde in Section2. We fix a branch of a factor z c (0) by cutting the Riemann sphere along R + - the positive part of real axis connecting z = 0 and z = ∞ . With this cut, ψ ( z ) and ψ ∞ ( z ) become single valued functions in their domains.The solutions ψ ( z ) and ψ ∞ ( z ) can be obtained as cohomological limits of the corresponding solutions in K-theory. Explicitly, let Ψ ( t , t , q, z ),Ψ ∞ ( t , t , q, z ) be the solutions of the K-theoretic QDE described in Section 4.Then lim τ → Ψ , ∞ ( e πi(cid:15) τ , e πi(cid:15) τ , e − πiτ , z ) = ψ , ∞ ( z )It follows that the transport of the qde:Tran( s ) := ψ ( e πis ) − ψ ∞ ( e πis ) (96)49s a limit the monodromy (51):Tran( s ) = lim τ → Mon ( z = e πis , t = e πi(cid:15) τ , t = e πi(cid:15) τ , q = e − πiτ ) (97)The next two subsections are devoted to the computation of this limit. ∇ DT Let us consider the transport of the fundamental solutions (31):Tran DT ( s ) = ψ DT ( e πis ) − ψ ∞ DT ( e πis ) . This transport matrix is related to (96) via conjugation by Γ DT . Proposition 11.
Tran DT ( s ) = lim τ → (cid:93) Mon ( e πis , e πi(cid:15) τ , e πi(cid:15) τ , e − πiτ ) (98) where (cid:93) Mon ( z, t , t , q ) := e ln( E
0) ln( z )ln( q ) U ( a, z ) − L ( a − , z − (cid:126) − ) e − ln( E ∞ ) ln( z )ln( q ) . (99) Proof.
By Corollary 1 and (52) we have
Mon ( z, t , t , q ) = Φ( q T X ) (cid:93) Mon ( z, t , t , q ) Φ( q T X ) − where (cid:93) Mon ( z, t , t , q ) is given by (99). The conjugation by Φ( q T X ) multi-plies the ( λ, µ )-matrix element of a matrix by F ( t , t , q ) = Φ( qT λ X )Φ( qT µ X ) . X is a symplectic variety, with the T -weight of the symplectic form given by (cid:126) . Thus, for a T -weight a of the tangent space T λ X there is the symplecticdual T -weight of T λ X given by a − (cid:126) − . This means that the above functionis of the form F ( t , t , q ) = (cid:89) ϕ ( aq ) ϕ ( qa − (cid:126) − ) ϕ ( qb ) ϕ ( qb − (cid:126) − )for some weights a and b . By Lemma 8 below we havelim τ → F ( e πiτ(cid:15) , e πiτ(cid:15) , e πiτ ) = Γ λ ( (cid:15) , (cid:15) )Γ µ ( (cid:15) , (cid:15) )50here Γ λ ( (cid:15) , (cid:15) ) = (cid:89) w ∈ char T ( T λ X ) w + 1) . (100)which are exactly the eigenvalues of the diagonal matrix Γ DT . We concludethat the limits in (97) and in (98) are related via the conjugation by Γ DT .The proposition follows. Lemma 8.
Let us consider a function of the form f ( a, b, (cid:126) , q ) = ϕ ( aq ) ϕ ( qa − (cid:126) − ) ϕ ( qb ) ϕ ( qb − (cid:126) − ) , then, for generic values of α, β, h we have lim τ → f ( e πiτα , e πiτβ , e πiτh , e πiτ ) = Γ( β + 1)Γ( − β − h + 1)Γ( α + 1)Γ( − α − h + 1) . (101) Proof.
Explicitly, we have: f ( a, b, (cid:126) , q ) = ∞ (cid:89) n =0 (1 − aq n +1 )(1 − a − (cid:126) − q n +1 )(1 − bq n +1 )(1 − b − (cid:126) − q n +1 ) , and formally computing the limit (101) we obtainlim τ → f ( e πiτα , e πiτβ , e πiτh , e πiτ ) = ∞ (cid:89) n =0 ( α + 1 + n )( − α − h + 1 + n )( β + 1 + n )( − β − h + 1 + n ) , (102)were we assume that the last infinite product exists. To show the convergenceof the product, we recall the Weierstrass product expansion of the Gammafunction: Γ( x ) = e − γx x ∞ (cid:89) n =1 (cid:16) xn (cid:17) − e x/n , which holds for all complex x except non-positive integers. Using this for-mula, after elementary manipulations, for generic α, β and h we obtainΓ( β + 1)Γ( − β − h + 1)Γ( α + 1)Γ( − α − h + 1) = ∞ (cid:89) n =0 ( α + 1 + n )( − α − h + 1 + n )( β + 1 + n )( − β − h + 1 + n ) . This proves that the product (102) exists for generic values of the parametersand also proves proves (101). 51 .2 Cohomological limits of balanced functions
Let us denote z = e πiξ , q = e πiτ , (cid:126) = e πih , and consider the Jacobi theta function ϑ ( ξ, τ ) = (cid:88) n ∈ Z e πin τ +2 πinξ = (cid:88) n ∈ Z q n / z n . This function is related to the q -theta function (56) via ϑ (cid:16) ξ − τ , τ (cid:17) = − z / ϕ ( q ) θ ( z ) . (103)The modular transformation of ϑ ( ξ, τ ) is described by the formula ϑ ( ξ, τ ) = i √ τ e − πiξ τ ϑ (cid:16) ξτ , − τ (cid:17) . (104) Lemma 9.
Let us consider the function ˜ f ( z, a, b, q ) = e m ln( z ) ln( a/b )ln( q ) f ( z, a, b, q ) , f ( z, a, b, q ) = θ ( z m a ) θ ( z m b ) with m ∈ Z . Then, for generic s ∈ R , the limit lim τ → ˜ f ( e πis , e πi(cid:15) τ , e πi(cid:15) τ , e − πiτ ) as τ approaches along the positive part of the imaginary axis, exists and isequal to the limit lim q → f ( q s , e πi(cid:15) , e πi(cid:15) , q ) . Proof.
For simplicity, let us consider a function of the form˜ f ( z, (cid:126) , q ) = e ln( z ) ln( (cid:126) )ln( q ) θ ( z (cid:126) ) θ ( z )i.e., we consider (107) m = 1, a = (cid:126) , b = 1. For the general values thecalculation is the same. Then˜ f ( e πis , e πihτ , e − πiτ ) = e − πish θ ( e πi ( s − hτ ) ) θ ( e πis ) . s is generic, we may assume that s (cid:54)∈ Z , and θ ( e πis ) (cid:54) = 0.Applying the modular transform using (103) and (104) we find θ ( e πi ( s − hτ ) ) θ ( e πis ) = e πih ( s +1 / ϑ (cid:16) sτ − h − + τ , − τ (cid:17) ϑ (cid:16) sτ − + τ , − τ (cid:17) . Let us denote new modular parameter by q = e − πi/τ . If τ → q →
0. Thuslim τ → ϑ (cid:16) sτ − h − + τ , − τ (cid:17) ϑ (cid:16) sτ − + τ , − τ (cid:17) = e − πih lim q → θ ( (cid:126) q s ) θ ( q s ) . Combining all these terms together we findlim τ → ˜ f ( e πiz , e πihτ , e − πiτ ) = lim q → θ ( (cid:126) q s ) θ ( q s ) = lim q → f ( q s , (cid:126) , q ) . This lemma relates τ → q → Proposition 12.
For s ∈ R \ Walls n , the transport of the quantum differentialequation is determined by the following limit: Tran DT ( s ) = lim q → R Ell ( q s , e πi(cid:15) , e πi(cid:15) , q ) . (105) where R Ell ( z, t , t , q ) = U ( a, z ) − L ( a − , z − (cid:126) − ) . Proof.
We have (cid:93)
Mon ( z, t , t , q ) := e ln( E
0) ln( z )ln( q ) R Ell ( z, t , t , q ) e − ln( E ∞ ) ln( z )ln( q ) . Recall that the matrix elements of U ( a, z ) and L ( a, z ) are balanced in z , seeDefinition 1 in [14]. This means that they depend on z through a combinationof theta functions of the form θ ( z m a ) θ ( z m b ) , m ∈ Z (106)53here a and b denote monomials in the equivariant parameters t , t . For theHilbert scheme X we know these functions explicitly [33].We see that the matrix elements of R Ell ( z, t , t , q ) are also balanced in z ,and the matrix elements of the monodromy matrix (cid:93) Mon ( z, t , t , q ) dependon z via combinations (106) and exponential factors of the form e ln( a ) ln( z )ln( q ) where a denotes a monomial in t , t . As (cid:93) Mon ( z, t , t , q ) is a q -periodicfunction in z , these exponential factors complete factors (106) to q -periodicfunctions. In other words, the matrix elements of (cid:93) Mon ( z, t , t , q ) only on z via the functions of the form˜ f ( z, t , t , q ) = e m ln( z ) ln( a/b )ln( q ) θ ( z m a ) θ ( z m b ) , (107)so that ˜ f ( zq, t , t , q ) = ˜ f ( z, t , t , q ).Let us consider limit (98) of (cid:93) Mon ( z, t , t , q ) with τ approaching 0 alongthe positive part of the imaginary axis. By Lemma 9 this limit exists and isequal to (105) for generic values of s .Finally, arguing as in the proof of Lemma 9, we see that the limits existsfor those s ∈ R with ms (cid:54)∈ Z , for all m appearing as exponents in (107)for all matrix elements of (cid:93) Mon ( z, t , t , q ). From the explicit formulas forthe elliptic stable envelope classes [33] we know that the exponents for X =Hilb n ( C ) are bounded by 0 < m ≤ n . This means that the limit exists forall s ∈ R \ Walls n .Combining Proposition 12 with results of Theorems 5 and 11 we obtain arepresentation-theoretic (in terms of B w ∈ U (cid:126) ( (cid:98)(cid:98) gl )) and an algebro-geometric(in terms of the K-theoretic stable envelopes) descriptions of the transport. Theorem 15.
The transport of the quantum connection ∇ DT from z = 0 to z = ∞ along a line γ s intersecting | z | = 1 at a non-singular point z = e πis equals Tran DT ( s ) = ←− (cid:81) w ∈ (0 ,s ) ( B • w ) − · T s ≥ , −→ (cid:81) w ∈ ( s, B • w · T , s < . hese matrices have the following Gauss decomposition Tran DT ( s ) = U ± s ( a ) − D s ( (cid:126) ) L ±− s ( a − ) where D s ( (cid:126) ) is diagonal matrix (81) and U ± s ( a ) , L ± s ( a ) denote the matricesof expansions of the K-theoretic stable envelope classes in the basis of fixedpoints (76). Remark 17.
Note that in the above theorem s (cid:54)∈ Walls n and thus U + s ( a ) = U − s ( a ) , L + s ( a ) = L − s ( a ) . Remark 18.
We need to clarify the ambiguity for choosing a path γ s forvalues of s which differ by 2 π . Let (cid:98) s (cid:99) the the integral part of s . Then γ s ishomotopy equivalent to a path which winds around z = 0 counterclockwise (cid:98) s (cid:99) times and then proceeds to z = ∞ intersecting | z | = 1 at e πis . Fig. 3below illustrates this situation. As z = 0 is a singular point of qde, windingaround this point amounts in a non-trivial monodromy and thus Tran DT ( s ) (cid:54) =Tran DT ( s + 2 π ).Figure 3: Example of two paths γ w , γ w with (cid:98) w (cid:99) = 0 and (cid:98) w (cid:99) = 2. s = 0 For s = 0 Theorem 15 gives Tran DT (0) = T . H ∗ = P ∗ and wearrive at the following result: Theorem 16. • The operator M ( ∞ ) is diagonal in the basis of Macdonald polynomialswith substitution P ∗ λ = P λ | p i = − p i . • The transport of the connection ∇ DT from z = 0 to z = ∞ along R + maps the eigenvectors of the operator M (0) to the eigenvectors of theoperator M ( ∞ ) . π ( P \ Sing , + ) In this section we compute the representation of the fundamental group π ( P \ Sing , z = 0 is also a singularity of the quantum differen-tial equation we consider the fundamental group π ( P \ Sing , + ) where 0 + is a point infinitesimally close to z = 0. More specifically (the relations inthe group will depend on this choice) we choose 0 + above the cut along R + ,see Fig. 4. This group is well defined.We choose the following generators of this group: • γ is a counterclockwise oriented loop based at 0 + around the singular-ity z = 0, • γ ∞ is a clockwise oriented loop based at 0 + around z = ∞ chosen asfollows: first is goes from 0 + to ∞ along the R + , then around z = ∞ and goes back to 0 + along R + . • γ w for w = ab is a counterclockwise oriented loop based at 0 + aroundsingularity located at the root of unity ζ = e πiab which do not intersect R + . It is clear that these loops are labeled by w ∈ Walls n with 0 < w < π ( P \ Sing , + ) is isomorphic to thefollowing group: (cid:68) γ , γ w , γ ∞ , w ∈ Walls n , < w < (cid:69)(cid:46)(cid:16) γ ←− (cid:89) w ∈ (0 , γ w = γ ∞ (cid:17) n = 3 this loop are shown in Fig 4. The relation satisfied by γ w isexplained in Fig 5.Figure 4: Generators of π ( P \ Sing , + ) for n = 3. Example 4.
For n = 3 the examples of compositions of generators of π ( P \ Sing , + ) for n = 3:Figure 5: Generators of π ( P \ Sing , + ) for n = 3.It is clear that the loop representing γ γ γ γ , in the above figure, is homo-topy equivalent to γ ∞ in Fig. 4 which gives the relation in the fundamentalgroup γ γ γ γ = γ ∞ . .5 Fock representation of the fundamental group The transformation properties of ψ DT ( z ) in the neighborhood of z = 0 arecompletely determined by the factor z c (0) in (19). Going around γ amountsin the transformation ψ DT ( γ · z ) = ψ DT ( z ) E where E is the diagonal matrix with eigenvalues e πic (0) λ = O (1) | λ = (cid:89) ( i,j ) ∈ λ t i − t j − . Thus ∇ DT ( γ ) = E . Similarly, the fundamental solution of the qde in the neighborhood of z = ∞ transforms as ψ ∞ DT ( z ) → ψ ∞ DT ( z ) E ∞ if we go around a small loop containing z = ∞ . This means that ∇ DT ( γ ∞ ) = T E ∞ T − where T = Tran DT (0) denotes the transport of ∇ DT from z = 0 to z = ∞ along R + . Theorem 4 describes the matrix T explicitly. Proposition 13.
For w ∈ Walls n with < w < we have ∇ DT ( γ w ) = B • w where B • w is the matrix of the operator (36) in the basis of the Macdonaldpolynomials.Proof. Let us consider s (cid:48) , s ∈ R so that there is only one element w ∈ Walls n such that 1 > s (cid:48) > w > s >
0. Then, by definition, ∇ DT ( γ w ) = Tran( s ) Tran( s (cid:48) ) − , The proposition follows from Theorem 15.Combining all this together we arrive at the following result
Theorem 17.
The representation (23) in generators is given by ∇ DT : γ → E , γ ∞ → T E ∞ T − , γ w → B • w . .6 Relation in π ( P \ Sing , + ) The generators of π ( P \ Sing , + ) are subject to the following relation γ ←− (cid:89) w ∈ (0 , γ w = γ ∞ . Applying the anti-homomorphism ∇ DT we obtain (cid:16) −→ (cid:89) w ∈ (0 , B • w (cid:17) E = T E ∞ T − . The left side of this equation is simply the matrix of the operator M ( ∞ )in the basis of fixed points. Thus, the last relation says that T maps theeigenvectors of M (0) to the eigenvectors of M ( ∞ ) which gives another proofof Theorem 16. Let j ( β ) λ denote the standard Jack polynomial with parameter β . We denote n βλ = (cid:89) (cid:3) ∈ λ ( l λ ( (cid:3) ) β − + a λ ( (cid:3) ) + 1)where a λ ( (cid:3) ) = λ i − j, l λ ( (cid:3) ) = λ (cid:48) j − i denote the standard arm and leg lengths of a box (cid:3) = ( i, j ) in λ and λ (cid:48) isthe transposed Young diagram. Let j ( β ) λ = n βλ j ( β ) λ and J λ = ( − (cid:15) ) | λ | j ( − (cid:15) /(cid:15) ) λ (cid:12)(cid:12)(cid:12) p i = − (cid:15) p i . The Jack polynomials normalized in this way correspond to the basis of torusfixed points in equivariant cohomology of Hilb n ( C ) under isomorphism (11).These polynomials form an eigenbasis of m (0).59ere are the first several examples of Jack polynomials normalized thisway: n = 1: J [1] = (cid:15) (cid:15) p n = 2: J [2] = (cid:15) (cid:15) p + (cid:15) (cid:15) p , J [1 , = (cid:15) (cid:15) p + (cid:15) (cid:15) p n = 3: J [3] = (cid:15) (cid:15) p + 3 (cid:15) (cid:15) p p + 2 (cid:15) (cid:15) p J [2 , = (cid:15) (cid:15) p + ( (cid:15) + (cid:15) ) (cid:15) (cid:15) p p + (cid:15) p (cid:15) J [1 , , = (cid:15) (cid:15) p + 3 (cid:15) (cid:15) p p + 2 (cid:15) (cid:15) p The transition matrix from the basis p λ to the basis of J λ has the form: n = 2: J = (cid:34) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:35) n = 3: J = (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) ( (cid:15) + (cid:15) ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) These matrices provide initial values of the fundamental solutions of thequantum differential equations normalized as in (20).
B Macdonald polynomials
Here is the first several polynomials in the normalization we use (see Section2.2 in [12] for definitions): n = 1: P [1] = p .n = 2: P [1 , = (1 + t ) p t + ( t − p t ,P [2] = (1 + t ) p t + ( t − p t . P , by definition, is such that its λ -th column is a vector P λ inbasis p µ . From above formulas we find: P = (cid:34) t t t t t − t t − t (cid:35) We compute ( − l = (cid:34) − (cid:35) , S = (cid:34) (cid:35) and thus P ∗ := ( − l P S = (cid:34) / t t / t t − / t − t − / t − t (cid:35) .n = 3: P [1 , , = ( t + t + 1) (1 + t ) p t + ( t −
1) ( t + t + 1) p p t + ( t − (1 + t ) p t ,P [2 , = ( t t + 2 t + 2 t + 1) p t t + ( t t − p p t t + ( t −
1) ( t − p t t ,P [3] = (1 + t ) ( t + t + 1) p t + ( t −
1) ( t + t + 1) p p t + ( t − (1 + t ) p t . P = ( t + t +1 ) (1+ t )6 t t t +2 t +2 t +16 t t (1+ t ) ( t + t +1 ) t ( t − ( t + t +1 ) t t t − t t ( t − ( t + t +1 ) t ( t − (1+ t )3 t ( t − t − t t ( t − (1+ t )3 t We compute ( − l = − , S = (108)and thus P ∗ = ( − l P S = / (1+ t ) ( t + t +1 ) t / t t +2 t +2 t +1 t t / t + t +1 ) (1+ t ) t − / ( t − ( t + t +1 ) t − / t t − t t − / ( t − ( t + t +1 ) t / ( t − (1+ t ) t / ( t − t − t t / ( t − (1+ t ) t Stable envelopes of X In this section we give explicit formulas for the matrices of K-theoretic stableenvelopes defined by (76). All formulas are written in variables: t = ah, t = ha . (109)so that h = (cid:126) / . n = 2: U +0 ( a ) = U − / ( a ) = (cid:34) ( − h + a ) ( a + h ) h a − ( h − h +1)( − h + a ) h a ( − h + a )( a +1)( a − a h (cid:35) U +1 / ( a ) = U − ( a ) = (cid:34) ( − h + a ) ( a + h ) h a − a ( − h + a )( h − h +1) h ( − h + a )( a +1)( a − a h (cid:35) By Lemma 6, L ± w ( a ) = SU ± w ( a − ) S and we compute: L +0 ( a ) = L − / ( a ) = (cid:34) ( ah − a +1)( a − ah ( h +1)( h − ah − h ( ah − ( ah +1) a h (cid:35) , L +1 / ( a ) = L − ( a ) = (cid:34) ( ah − a +1)( a − ah ( h +1)( h − ah − a h ( ah − ( ah +1) a h (cid:35) . All other stable envelope matrices differ from the ones given above by anintegral shift of the slope, and can be computed by: U ± w +1 ( a ) = O (1) U ± w ( a ) O (1) − , L ± w +1 ( a ) = O (1) L ± w ( a ) O (1) − , with O (1) = (cid:34) ah ah (cid:35) .n = 3: U +0 ( a ) = U − / ( a ) = − ( − h + a ) ( a + h ) ( a + ah + h ) a / h / ( − h + a ) ( h − h +1)( a + h ) h / a / − ( − h + a )( h − h +1) ( ah − ) a / h / − ( − h + a ) ( a − h ) h / a / ( ah +1)( − h + a )( h − h +1)( a +1)( a − a / h / − ( a +1)( a − ( a h − ) ( − h + a ) h / a / +1 / ( a ) = U − / ( a ) = − ( − h + a ) ( a + h ) ( a + ah + h ) a / h / ( h − h +1)( − h + a ) ( a +1 ) h / √ a − ( h − h +1)( − h + a ) ( a h − a h − a + h ) a / h / − ( − h + a ) ( a − h ) h / a / ( a +1 ) ( − h + a )( h − h +1)( a − a +1) h / a / − ( a +1)( a − ( a h − ) ( − h + a ) h / a / U +1 / ( a ) = U − / ( a ) = − ( − h + a ) ( a + h ) ( a + ah + h ) a / h / ( h − h +1)( − h + a ) ( a +1 ) h / √ a ( h − h +1)( − h + a ) ( a h − a h − a h +1 ) h / √ a − ( − h + a ) ( a − h ) h / a / ( a +1 ) ( − h + a )( h − h +1)( a − a +1) h / a / − ( a +1)( a − ( a h − ) ( − h + a ) h / a / U +2 / ( a ) = U − ( a ) = − ( − h + a ) ( a + h ) ( a + ah + h ) a / h / ( − h + a ) a / ( h − h +1)( a + h ) h / − ( h − h +1) a / ( − h + a ) ( ah − ) h / − ( − h + a ) ( a − h ) h / a / ( ah +1)( − h + a )( h − h +1)( a +1)( a − √ ah / − ( a +1)( a − ( a h − ) ( − h + a ) h / a / By Lemma 6, L ± w ( a ) = SU ± w ( a − ) S and using (108) these matrices are easyto compute explicitly. The stable envelope matrices for other slopes aredetermined from U ± w +1 ( a ) = O (1) U ± w ( a ) O (1) − , L ± w +1 ( a ) = O (1) L ± w ( a ) O (1) − , with O (1) = a h h −
00 0 a h . Stable envelopes for Y w Here we give examples of stable envelope matrices (77):Case n = 2: U + Y ( a ) = (cid:34) ( a + hh )( − hh + a ) hh a − ( hh +1)( hh − − hh + a ) hh a ( − hh + a )( a +1)( a − a hh (cid:35) U + Y / ( a ) = − ( − hh + a )( a + hh ) √ a hh / √ a ( hh − hh +1) hh / − ( a +1)( a − √ hh a / U − Y ( a ) = (cid:34) ( a + hh )( − hh + a ) hh a − ( hh +1)( hh − − hh + a ) hh a ( − hh + a )( a +1)( a − a hh (cid:35) U − Y / ( a ) = − ( − hh + a )( a + hh ) √ a hh / ( hh − hh +1) a / hh / − ( a +1)( a − √ hh a / By Lemma 6 L ± Y w ( a ) = S U ± Y w ( a − ) S and we compute: L + Y ( a ) = (cid:34) ( a hh − a +1)( a − a hh ( a hh − hh − hh +1) hh ( a hh − ( a hh +1) a hh (cid:35) L + Y / ( a ) = ( a +1)( a − √ a √ hh ( hh − hh +1) √ a hh / ( a hh − a hh +1) a / hh / L − Y ( a ) = (cid:34) ( a hh − a +1)( a − a hh ( a hh − hh − hh +1) hh ( a hh − ( a hh +1) a hh (cid:35) L − Y / ( a ) = ( a +1)( a − √ a √ hh ( hh − hh +1) a / hh / ( a hh − a hh +1) a / hh / n = 3: U + Y ( a ) = − ( a + hh )( − hh + a ) ( a + a hh + hh ) a / hh / ( hh − hh +1)( − hh + a ) ( a + hh ) hh / a / − ( a hh − ) ( hh +1)( hh − − hh + a ) a / hh / − ( − hh + a ) ( a − hh ) hh / a / ( a hh +1)( − hh + a )( a +1)( a − hh − hh +1) a / hh / − ( − hh + a )( a +1)( a − ( a hh − ) hh / a / U + Y / ( a ) = − √ a ( − hh + a )( a + hh ) hh / ( hh − hh +1) hh / √ a hh −
00 0 − ( a +1)( a − hh / a / U + Y / ( a ) = − ( − hh + a ) ( a + a hh + hh ) hh ( hh − hh +1) a / hh / ( hh − hh +1) hh − a − hh a / hh / ( hh − hh +1) hh − a hh − a hh − Y ( a ) = − ( a + hh )( − hh + a ) ( a + a hh + hh ) a / hh / ( hh − hh +1)( − hh + a ) ( a + hh ) hh / a / − ( a hh − ) ( hh +1)( hh − − hh + a ) a / hh / − ( − hh + a ) ( a − hh ) hh / a / ( a hh +1)( − hh + a )( a +1)( a − hh − hh +1) a / hh / − ( − hh + a )( a +1)( a − ( a hh − ) hh / a / U − Y / ( a ) = − √ a ( − hh + a )( a + hh ) hh / ( hh − hh +1) hh / a / hh −
00 0 − ( a +1)( a − hh / a / U − Y / ( a ) = − ( − hh + a ) ( a + a hh + hh ) hh ( hh − hh +1) a / hh / ( hh − hh +1) a hh − a − hh a / hh / ( hh − hh +1) a hh − a hh − a hh By Lemma 6, L ± Y w ( a ) = S U ± Y w ( a − ) S and the matrices L ± Y w ( a ) are easy tocompute explicitly using (108).If w = a/b, w (cid:48) = a (cid:48) /b (cid:48) with the same denominator b = b (cid:48) then Y w = Y w (cid:48) .Thus, in cases n = 2 and n = 3 the matrices U ± Y w ( a ), L ± Y w ( a ) are equal to oneof listed above. E Twisted R-matrices for Y w The R-matrices of Y w are defined by (82). Using explicit matrices from theprevious section we compute: n = 2: R + Y ( a ) = ( − h + a )( a +1)( a − h ( ah − ( ah +1) − ( h +1)( h − − h + a ) a ( ah − ( ah +1) − ( h +1)( h − − h + a ) a ( ah − ( ah +1) ( − h + a )( a +1)( a − h ( ah − ( ah +1) R − Y ( a ) = ( − h + a )( a +1)( a − h ( ah − ( ah +1) − ( h +1)( h − − h + a ) a ( ah − ( ah +1) − ( h +1)( h − − h + a ) a ( ah − ( ah +1) ( − h + a )( a +1)( a − h ( ah − ( ah +1) + Y / ( a ) = − ( a +1)( a − h ( ah − ah +1) a ( h − h +1)( ah − ah +1)( h +1)( h − ah − ah +1) − ( a +1)( a − h ( ah − ah +1) R − Y / ( a ) = − ( a +1)( a − h ( ah − ah +1) ( h +1)( h − ah − ah +1) a ( h − h +1)( ah − ah +1) − ( a +1)( a − h ( ah − ah +1) n = 3: R + Y ( a ) = − ( a +1)( a − ( a h − ) ( − h + a ) h ( ah − ( ah +1)( a h + ah +1) ( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( − h + a )( h − h +1) ( ah − ) a ( ah − ( ah +1)( a h + ah +1)( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( a h − a h + a h − a h + a − ah + h ) ( − h + a )( ah − ( a h + ah +1) ( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( − h + a )( h − h +1) ( ah − ) a ( ah − ( ah +1)( a h + ah +1) ( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( a +1)( a − ( a h − ) ( − h + a ) h ( ah − ( ah +1)( a h + ah +1) R + Y / ( a ) = − h ( a h − ) ( ah − a h + ah +1) ( h +1)( h − ha ( ah − a h + ah +1) ( h +1)( h − a ( ah − a h + ah +1)( h +1)( h − ah − a h + ah +1) − h ( a h − ) ( ah − a h + ah +1) ( h +1)( h − ha ( ah − a h + ah +1)( h +1)( h − h ( ah − a h + ah +1) ( h +1)( h − ah − a h + ah +1) − h ( a h − ) ( ah − a h + ah +1) R + Y / ( a ) = − ( a +1)( a − h ( ah − ah +1) a ( h − h +1)( ah − ah +1) ( h +1)( h − ah − ah +1) − ( a +1)( a − h ( ah − ah +1) R − Y ( a ) = − ( a +1)( a − ( a h − ) ( − h + a ) h ( ah − ( ah +1)( a h + ah +1) ( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( − h + a )( h − h +1) ( ah − ) a ( ah − ( ah +1)( a h + ah +1)( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( a h − a h + a h − a h + a − ah + h ) ( − h + a )( ah − ( a h + ah +1) ( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( − h + a )( h − h +1) ( ah − ) a ( ah − ( ah +1)( a h + ah +1) ( h +1)( h − − h + a )( a − a +1) ah ( ah − ( a h + ah +1) − ( a +1)( a − ( a h − ) ( − h + a ) h ( ah − ( ah +1)( a h + ah +1) − Y / ( a ) = − h ( a h − ) ( ah − a h + ah +1) ( h +1)( h − ah − a h + ah +1) ( h +1)( h − h ( ah − a h + ah +1)( h +1)( h − ha ( ah − a h + ah +1) − h ( a h − ) ( ah − a h + ah +1) ( h +1)( h − ah − a h + ah +1)( h +1)( h − a ( ah − a h + ah +1) ( h +1)( h − ha ( ah − a h + ah +1) − h ( a h − ) ( ah − a h + ah +1) R − Y / ( a ) = − ( a +1)( a − h ( ah − ah +1) ( h +1)( h − ah − ah +1) a ( h − h +1)( ah − ah +1) − ( a +1)( a − h ( ah − ah +1) Using (82) one also computes similar matrices R + Y / ( a ). It might be instruc-tive to check Proposition 9 which in this case claims that R + Y / ( a − ) R − Y / ( a ) = 1 (110)The matrix γ w ( a ) defined by (78). Here are the first examples: γ ( a ) = (cid:34) − h − − h − (cid:35) , γ / ( a ) = (cid:34) √− ah h √− a (cid:35) γ ( a ) = ( − h − ) / − h √− h −
00 0 ( − h − ) / ,γ / ( a ) = − ( − h − ) / a h √− h −
00 0 − a ( − h − ) / γ / ( a ) = − ( − h − ) / a √− a h √− h −
00 0 − a √− a ( − h − ) / / ( a ) = ( − h − ) / a − h √− h −
00 0 a ( − h − ) / Let κ ∗ denote the substitution (63): κ ∗ : a → zh, h → h − , z → ah The twisted R -matrices are defined by (83). Using the above explicit formulaswe find the first several examples: n = 2: R − Y ( z ) = ( zh +1)( zh − ( h z − ) h ( z − ( z +1) z ( h +1)( h − ( h z − ) h ( z − ( z +1) z ( h +1)( h − ( h z − ) h ( z − ( z +1) ( zh +1)( zh − ( h z − ) h ( z − ( z +1) R − Y / ( z ) = − h z − h ( z − a ( h − ) h ( z − z ( h − ) a ( z − − h z − h ( z − n = 3: R − Y ( z ) = − ( h z − )( h z − ) ( zh − zh +1) h ( z − ( z +1)( z + z +1) − z ( zh +1)( zh − h +1)( h − ( h z − ) h ( z − ( z + z +1) − ( h − h +1) ( h z − )( h − z ) z h ( z − ( z +1)( z + z +1) − z ( zh +1)( zh − h +1)( h − ( h z − ) h ( z − ( z + z +1) − ( h z + h z − h z − h z − h z + z +1 )( h z − ) h ( z − ( z + z +1) − z ( zh +1)( zh − h +1)( h − ( h z − ) h ( z − ( z + z +1) − ( h − h +1) ( h z − )( h − z ) z h ( z − ( z +1)( z + z +1) − z ( zh +1)( zh − h +1)( h − ( h z − ) h ( z − ( z + z +1) − ( h z − )( h z − ) ( zh − zh +1) h ( z − ( z +1)( z + z +1) R − Y / ( z ) = − h z − h ( z − a ( h − ) h ( z − − a ( h − ) h ( z − z ( h − ) a ( z − − h z − h ( z − a ( h − ) h ( z − − hz ( h − ) a ( z − z ( h − ) a ( z − − h z − h ( z − R − Y / ( z ) = − h z − h ( z − a ( h − ) h ( z − z ( h − ) a ( z − − h z − h ( z − − Y / ( z ) = − h z − h ( z − a ( h − ) h ( z − − a ( h − ) h ( z − hz ( h − ) a ( z − − h z − h ( z − a ( h − ) h ( z − − hz ( h − ) a ( z − z ( h − ) ha ( z − − h z − h ( z − Note that the properties of these matrices are in full agreement with Propo-sition 8. In particular, their matrix elements are monomials in a . F Monodromy and wall-crossing operators
By Theorem 12 (cid:121) B w ( z ) = (cid:126) Ω w R − Y w ( z ) . the twisted R -matrices R − Y w ( z ) are computed in the previous section and theoperators (cid:126) Ω w are given by (87). We find: n = 2: (cid:126) Ω = (cid:34) h h (cid:35) , (cid:126) Ω / = (cid:34) − h − h (cid:35) n = 3: (cid:126) Ω = − h − h
00 0 − h , (cid:126) Ω / = − h − h
00 0 − h , (cid:126) Ω / = − h − h , (cid:126) Ω / = − h − h
00 0 − h . The wall crossing operators in the basis of fixed points (i.e., Macdonaldpolynomials) can be computed using explicit matrices from Appendix C by B • w ( z ) = U − w ( a ) − (cid:126) Ω w R − Y w ( z ) U + w ( a )70or instance, in the case n = 2 we find: B • ( z ) = ( h z − )( a h z + ah z − h z − ah z − a h + az + h ) h ( z − ( a +1)( a − z +1) ( h − h +1)( ah +1)( ah − ( h z − ) zah ( z − ( a +1)( a − z +1) a ( h − h +1)( − h + a )( a + h ) ( h z − ) zh ( z − ( a +1)( a − z +1) ( h z − )( a h z − ah z − h z +2 ah z − a h − az + h ) h ( z − ( a +1)( a − z +1) B • / ( z ) = a h z − h z + h z − a − z +1( z − z +1)( a +1)( a − − ( ah − h − h +1) z ( ah +1)( z − z +1)( a +1)( a − − ( − h + a )( h − h +1) z ( a + h )( z − z +1)( a +1)( a − a h z − a h z + z a − h z − a +1( z − z +1)( a +1)( a − For n = 3 these matrices are already too large to print here. G Representation of the fundamental group
For n = 2, the fundamental group π ( P \ Sing , + ) is generated by γ , γ / , γ ∞ subject to a relation γ γ / = γ ∞ . By Theorem 17 ∇ DT ( γ ) = E , ∇ DT ( γ ∞ ) = T − E ∞ T , ∇ DT ( γ / ) = B • / and thus, we need to check that B • / E = T − E ∞ T . (111)From definitions, we compute E = (cid:34) ah ah (cid:35) , E ∞ = (cid:34) ah ha (cid:35) , B • w := B • / ( ∞ ) and thus from the matrix in the previous section we find: B • / = a h − h + h − a +1)( a − − ( ah − h − h +1)( ah +1)( a +1)( a − − ( − h + a )( h − h +1)( a + h )( a +1)( a − a h − a h + a − h ( a +1)( a − Finally, by definition T = P − P ∗ and from explicit matrices in Appendix B,in variables (109) we compute: T = ( ah +1)( ah − ah ( a +1)( a − a ( h − h +1)( a − a +1) h − a ( h − h +1)( a − a +1) h a ( a + h )( − h + a )( a − a +1) h , ∇ GW ( γ (cid:48) ) for loops based at z = 1. Using Proposition 1 we com-pute: ∇ GW ( γ (cid:48) ) = P E P − = (cid:34) a h + ah − a + h ah − ( ah +1)( a + h )2 ah − ( ah − − h + a )2 ah a h − ah + a + h ah (cid:35) , ∇ GW ( γ (cid:48) / ) = P B • / P − = (cid:34) − a h − ah − a h + h − a − h a − ( a + h )( h − h +1)( ah +1)2 a ( h +1)( h − ah − − h + a )2 a a h + ah − a h + h + a − h a (cid:35) , ∇ GW ( γ (cid:48)∞ ) = P T − E ∞ T P − = (cid:34) a h − ah + a + h a − ( ah +1)( a + h )2 a − ( ah − − h + a )2 a a h + ah − a + h a (cid:35) . We observe that, in full agreement with Theorem 2, all matrix of ∇ GW ( γ (cid:48) )are Laurent polynomials in a and h .Similarly, in the case n = 3, the fundamental group π ( P \ Sing , + ) isgenerated by γ , γ / , γ / , γ / , γ ∞ subject to the relation γ γ / γ / γ / = γ ∞ By Theorem 17 we need to check the relation B • / B • / B • / E = T − E ∞ T . This relation, and the polynomiality of ∇ GW ( γ (cid:48) ) are checked using explicitmatrices below: 72 • / = + ( a + h ) ( a + a h + h ) ( h + )( h − )( − h + a ) ( a + )( a − ) ( a − h ) − ( a h + a + h ) h ( h + )( h − )( a h − )( − h + a ) ( a + )( a − ) ( a − h ) ( a h + ) ( a h + a h + ) ( h + )( h − )( a h − ) a ( a − h ) ( a − )( a + ) − ( a + a h + ) ( a + h ) ( a + a h + h ) ( h + )( h − )( − h + a ) ( a − h )( a h − ) h ( h − )( h + )( − h + a ) ( a h + a + h )( a + a h + ) ( a h − ) ( a − h )( a h − ) − ( a + a h + ) ( a h + ) ( a h + a h + ) ( h + )( h − )( a h − ) a ( a − h )( a h − ) ( a + a h + h ) ( a + h ) a ( h + )( h − )( − h + a ) ( a + )( a − ) ( a h − ) − ( a h + a + h ) h a ( h + )( h − )( a h − )( − h + a ) ( a + )( a − ) ( a h − ) ( a h + ) ( a h + a h + ) ( h + )( h − )( a h − ) ( a + )( a − ) ( a h − ) B • / = + ( h − )( h + )( − h + a )( a + h )( a h − ) ( a + a h + h ) h ( a + )( a − ) ( a − h ) − a ( a + ) ( h + ) ( h − ) ( a h − ) ( a + )( a − ) ( a − h ) − ( a h + ) ( a h + a h + ) ( h + )( h − )( a h − ) h ( a + )( a − ) ( a − h ) − ( a + ) ( a + h ) ( a + a h + h ) ( h + ) ( h − ) ( − h + a ) h ( a − h )( a h − ) a ( h − ) ( h + ) ( a + ) ( a − h )( a h − )( a + ) ( a h + ) ( a h + a h + ) ( h + ) ( h − ) ( a h − ) h ( a − h )( a h − ) − ( a + h ) ( a + a h + h ) ( h + )( h − )( − h + a ) h ( a + )( a − ) ( a h − ) a ( a + ) ( h + ) ( h − ) ( − h + a ) ( a + )( a − ) ( a h − ) ( h − )( h + )( a h − )( − h + a )( a h + ) ( a h + a h + ) h ( a + )( a − ) ( a h − ) B • / = + ( a + h ) ( a + a h + h ) ( h + )( h − )( − h + a ) ( a + )( a − ) ( a − h ) − ( a + a h + ) h a ( h + )( h − )( a h − )( − h + a ) ( a + )( a − ) ( a − h )( a h + a h + ) ( a h + )( h + )( h − ) a ( a h − ) ( a + )( a − ) ( a − h ) − ( a h + a + h ) ( a + h ) ( a + a h + h ) ( h + )( h − )( − h + a ) a ( a − h )( a h − ) h ( h − )( h + )( − h + a ) ( a h + a + h )( a + a h + ) ( a h − ) ( a − h )( a h − ) − ( a h + a + h ) ( a h + ) ( a h + a h + ) ( h + )( h − )( a h − ) ( a − h )( a h − ) ( a + h ) ( a + a h + h ) ( h + )( h − )( − h + a ) a ( a + )( a − ) ( a h − ) − ( a + a h + ) h ( h + )( h − )( a h − )( − h + a ) ( a + )( a − ) ( a h − ) ( a h + ) ( a h + a h + ) ( h + )( h − )( a h − ) ( a + )( a − ) ( a h − ) T = ( a h − ) ( a h + a h + ) ( a h + ) h a ( a + )( a − ) ( a − h ) ( a h − )( h − )( h + )( a h + ) h ( a + )( a − ) ( a − h ) − a ( h − )( h + ) ( − h + a ) h ( a + )( a − ) ( a − h ) − ( h − )( h + ) ( a h + a h + ) ( a h + )( a h − )( a + h ) a h ( a − h )( a h − ) a h − a h − a h + a h − a h − a h + a h − a − a h + h h ( a h − )( a − h ) ( − h + a )( a + h ) ( a + a h + h ) ( a h + )( h − )( h + ) a h ( a − h )( a h − ) ( a h − ) ( h − )( h + ) a h ( a h − ) ( a + )( a − ) − a ( − h + a )( h − )( h + )( a + h ) h ( a + )( a − ) ( a h − )( a + a h + h ) ( a + h )( − h + a ) a h ( a h − ) ( a + )( a − ) E = a h
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